Blood magic is not truly forbidden. Not even in the highly conservative British wizarding society. Instead it is highly regulated. A mundane equivalent would be nuclear technology. Like nuclear technology, the uses and dangers of blood magic are vast and not fully understood. Because of this, legal practitioners of Blood magic submit themselves to a great deal of governmental regulation and oversite.
James and Lily Potter were magical prodigies and while neither worked in the field of Blood Magics, they both had a remarkable understanding of it in addition to other magics. So when it became clear that their son was a target of the worst Dark wizard in a millennia, they both devoted their intellects toward protecting their most prized achievement.
In the end, what Lily and James Potter did to protect their son was the equivalent of two very brilliant physicists building a thermonuclear bomb in their basement. To complete the analogy, the complexity of the particular blood magic ritual that the Potter's created and implemented would be the equivalent of those two physicists building a nuclear bomb in their basement and that bomb having all the yield of Hiroshima and Nagasaki combined with no radioactive fallout. Basically, the two magicals created what was previously thought impossible.
On arrival after the tragedy, Albus Dumbledore, himself considered the greatest and most knowledgeable magic user of the past 2 ages, was able to determine the result but unable to decipher how it was done. The only thing he was sure about was that one of the deathly hallows had been used as part of the ritual. The result was that Harry Potter's blood was granted a protection that was tied directly into his father's legacy and his mother's love.
The great wizard could not figure out how to use the Potter Legacy but was able to find that the home of anyone with Lily's blood could power near impregnable wards. Acting on this he quickly made arrangements and secreted Harry Potter off to his aunt's home.
James and Lily did not aim to protect their son's home, they aimed to protect their son and had Albus Dumbledore left their efforts alone the power of that protection would have ejected the leach that was currently attached to his forehead. Instead Albus Dumbledore had diverted some of the Potter's efforts toward warding his home leaving just enough power to contain but not eradicate the soul of the most evil and powerful dark lord in centuries.
…Smallest Bedroom, 4 Privet Dr., Surrey, England
Sleeping on his bed in Petunia's home, Harry Potter began to glow. The foreign soul now stood out in Harry's mind and magic due to his semi-accidental occlumency and the power of the protection ritual recognized it as the greatest threat to its charge. Before it had been focused on protecting Harry from his relatives which was why his abuse never became as physical as the Dursleys might have wanted.
Now, a combination of Harry's new life plus the clarity of the magical infection against his organize mind caused the protection to break from Dumbledore's meddling, collapsing the houses ward, and put the full force of that protective magic to use against the soul fragment.
Harry's scar burst open while he slept peacefully in a magically induced coma. Part of the original ritual's magic was designed to use the attacker's power against it and the soul fragment had much of Voldemort's power still in it, much more than was needed to destroy it. The ritual was designed to feed any excess power into the Potter legacy. If Harry was in a Potter Property or maybe even in a magical area the excess magic would have gone to a Potter property's wards or to a house-elf. But living in muggle England meant that Harry himself was the only example of the Potter Legacy anywhere close.
Protective magic cannot destroy what it is meant to protect and so the excess magically activated all of the latent magical abilities that existed within Harry's DNA, strengthened and increased his magical core, and ripped all the magical knowledge from the dying soul and dumped it into Harry's mind. With the last bit of excess magic the ritual anchored itself to Harry's mind and core in addition to his blood.
The collapse of the wards around Privet Drive would have been surprising to anyone familiar with wards. Wards are like a magical shield powered by a magical power transformer and collapsing them would generally result in a magical backlash. That backlash can usually be felt by anyone attuned to magic (even squibs). A very powerful ward, like the one on the Dursley home would even blink into the visual spectrum upon collapsing.
A ward monitor worked by being tied into the ward power and would self-destruct from the excess power backlash of a collapse. The ward monitor in Albus Dumbledore's office did not self-destruct. It just stopped. The reason this happened is because the ward did not collapse in the classic sense, a better description would be that it just lost power.
Typically the only way to depower a ward that is tied to someone's blood is to kill the person anchoring the ward. But like blood and bone, magic does not dissipate instantly on the death of an individual, it is a gradual process and so too would the depowering of any ward tied to it. That was why the look of fear and surprise and confusion that appeared on the Hogwarts headmaster's face when the blood ward on the Dursley residence depowered was one for the ages.
…Headmaster 's Office, Hogwarts School of Witchcraft and Wizardry
Moving like a man half his age Albus Dumbledore jumped up and called for his Phoenix, Fawkes, to take him directly to Harry Potter. What he found was a relief but equally confusing. A glowing Harry Potter was laying flatly on his back with a serene smile on his face. That serenity seemed in contrast with the wide tear streaks that went down his face. Fortunately, Albus would still have opportunity to solve those mysteries as the boy was obviously still breathing.
After a small sigh of relief, the wizard drew his wand and started the process to try and re-establish the blood wards. Unfortunately the ritual had stabilized and Dumbledore could no longer try and modify it. He had never truly figured out what Lily had done and had no clue that James was a part of the ritual at all. His inability to re-establish the ward lead him to check on Petunia, fearing that she had died or been replaced causing Lily's blood to no longer reside at the home.
He cast a sleeping spell on the other occupants of the house and checked Petunia, who was sleeping next to Vernon. He checked for all magical means to copy a person's identity and verified Petunia's blood as still being Lily's sister before returning to Harry's bedroom with a frown. The only thing he could consider was for Harry or Petunia to no longer consider this place their home, but magic was pretty literal when it came to these things and Harry or Petunia would have to establish another home and live in it before that could be true.
Looking around the room it was obvious that the boy lived in this room. His schoolbooks and a few of his toys were scattered about and the boy was dressed for bed and sleeping in said bed. Also he hadn't noticed before he left the castle but all of the remaining wards on the property were still intact. Only the blood ward had collapsed. Albus put aside his intellectual curiosity and refocused on what he would do now. His plans had recently taken a minor blow because of the intrusion of some local muggles. The boy's home life was no longer ideal to the man's plans and he was irritated that the Dursleys had been so bad that people took notice.
More devastating was that the muggles had created the bond that he hoped to create when he gave the boy a place to "escape" to. Magic was a powerful lure though and he would still be responsible for introducing the boy to it. When the time came he would still have Hagrid fetch the boy but would make sure he was in the alley. Being introduced by Hagrid while all the witches and wizards looked at him in awe would cement his position in the boys mind.
It wasn't as full proof as the "desperate orphan" plan but it should work. Albus was pulled from his thoughts and frowned at the glowing boy. The glowing would be a problem. His frown deepened as none of his detection spells would register against the foreign magic. He sent a mild stunner and was unsurprised to see it die at the outer edge of the glowing shield. He cursed himself for not binding the boy's magic.
After reflecting Tom's curse, Albus had arrived at the Potter home with intention of binding the powerful wizard only to find normal power levels and evidence that Lily had performed some sort of ritual. Leaving the boy unbound now seemed to be a mistake as the boy did indeed have some unknown power. As he wondered what to do, the boy's glow started to recede until it disappeared. "One problem down", he thought.
He sent another stunner and watched it die before hitting the boy. He pulled a small knife from his robes and tried to get a sample of blood but the knife would not pierce the child's skin. "What had happened here?" he thought before looking up at the boy's forehead and noticing that the scar had both grown and shrunk at the same time. It no longer looked swollen and perpetually irritated and instead was completely healed. But while it seemed completely healed, it had also grown and seemed more stylized.
As if someone had carved a lightning bolt into his head purposefully. It was obviously no longer a cursed scar and Albus wondered if his other suspicion had been purged from the location as well. Many of Albus' plans seemed to have died this night and he needed to regroup. He figured the situation was stable for now and he had some time to re-adjust. Plans already starting to form in his mind, Albus left and returned to Hogwarts.