It's been a while since you posted this query, but I see you've gotten no replies, so I thought I'd take a stab at it. If you have Ancestry.com, do a search on military records under the name "Andrew Baines." You didn't mention the date of the letter you found, but if 1946 is a viable date, Ancestry shows someone with that name serving from January through October 1946 on the USS Cape Gloucester, and in December 1946 on the USS Bougainville. By his service number, it appears all listings are for the same individual.
I tried Googling his name, but found there is a well-known person in the UK by that same name on the first several hits. However, with persisitence, you might find something there on the several pages of hits that come up.
Best wishes with your search. What an interesting find!