1) You can use the Finnair website to change your FF program and membership number. Have your Finnair one in there to start with, for seat selection and check in, then change it to the BA one before the flight (but after entering the lounge.) If you have another flight switch it back to Finnair before check in, then switch to BA before the flight. I haven't tried this but I have heard that others have and claimed it worked.
2) If you get a good agent on the phone, you can have them put in different programs and membership numbers for FQTS (used for status) and FQTV (used for earning miles/avios.) This is what I have done in the past (although not for quite a few years!)
3) You could just have your BA number in there and flash your Finnair card when necessary (not a particularly high tech method, but should work, mostly!) Although this won't get you seat selection, so in this case you might want to use a hybrid of #1 and #3 .
I found #2 pretty easy - one call (maybe two if you don't get the right agent first time) and it's done! (Assuming BA's IT systems still provide for this.)
rb211.