My personal opinion is that he should be buried in York, where he wished to be, and with a full Catholic funeral mass, at that. Whatever he was, Richard was England’s anointed king - anointed and crowned King at a grand, solemn and very well-attended ceremony at Westminster Abbey on July 6 1483, and he was of course a Catholic. He was also, in fact, austerely religious, a public benefactor and protector of the Church, a founder of charities, who throughout his life upheld a strict code of sexual morality, in marked contrast to many of his fellow courtiers. He was also an extremely popular King who some dearly loved; the preferences, then, of the large number of his collateral descendents should be respected, just as he respected others' wishes and opinions.
However, it seems likely that he will be buried in Leicester and according to the current formal religious rites of the Anglican Church. Whatever the final decision I hope that he is interred with full regal honours and as soon as possible to mitigate any further fruitless controversy attending the dignity of this good man, particularly as he wasn't only killed, he was brutalised in quite horrific ways and his body forced into a hole in the ground not large enough to accommodate it. This last is unheard of treatment of a King defeated in battle almost everywhere in the world, ever.