Hi CTEASDEL
I'd suggest calling the Diamond Club on 0870 606 99 77 and asking. I've never done this myself, but from previous posts on this board they're fairly good about it, particularly if you're moving from BA.
I'd suggest probably best not to mention your UA Gold to begin with. They might be a bit iffy about pinching members from other *A carriers (not that I'm suggesting you would be ditching your UAL membership). No basis for suggesting this, just a hunch. They should give it to you purely on the basis of your BA Gold. (You may need to fax statements and the like.)
Welcome to BMI! It's so much nicer that the competition...