A person who has a weird accent that you would expect. It's a good way of saying what you want to say without offending anybody.
Here's where it come from:
Way back when, the British colonized India. *duh* Anyway, since India so friggin' huge, different British communities popped up in different place. And it just so happens that the Welsh ended up in Bombay. Well, since there were so many Britons around, the locals ended up learning the language. So what you had were people that looked Indian, but spoke perfect English with, of all things, a Welsh accent. Hence the term Bombay.
By the way, I learned this from my mom. I love my mommy!
For the Yanks, it really weird for them to see a black person speaking with an English accent. To them, they're bombay welsh.
Comparing the way British Indians speak with the Welsh accent
Those new neighbours speak very good Bombay Welsh