The Books of Magic series undoubtedly had an influence on the Harry Potter series. Both series feature an English child who finds himself thrust into an already existing world of magic. Harry Potter looks remarkably similar to the protagonist Tim Hunter of Books of Magic. It is safe to assume J.K. Rowling received some inspiration from the Books of Magic series.
Both books stem from a pre-tween boy essentially at a crossroads in life, deciding whether to embrace the magical/wizard side of himself. While the actual worlds are distinct, the main idea is that they are split between the real world and the mystical and supernatural world. They both forge new communities within these different realms.
Yes, J.K. Rowling was influenced by Neil Gaiman's series. Tim Hunter and Harry Potter are very similar, with almost the same background story. They both find out they can do magic, have magic parentage and both start their training in the magical arts. Also, they both have an owl companion, and both boys have black hair, and are spectacled.
Neil Gaiman's Books of Magic was not an influence on J.K. Rowling as she was writing Harry Potter. Because it was a comic book, it is very unlikely that J.K Rowling read it because comics are not widely distributed. J.K. Rowling has said in interviews that she got the idea for Harry Potter when she was on a train and wishing she was going to a magical place, so her inspiration was not from Gaiman's Books of Magic.