Buddug = BITH ig
Begw = BEH goo (this is a short form of Margaret, a bit like Peggy. I've seen it a few times as a full name lately)
If it's pronounced DEL iss it might be a spin on Delyth, but I've never seen it in Wales, & Delyth is so unusual that I'd guess it was much more likely to be a spin on Denise.:)