Q1 Yes, they should be able to check them through. If they don't for some reason, go to the transfer desk airside and have your bags retagged without having to pick them up even if they don't check them all the way through. SIN is highly efficient in doing this even on totally separate tickets on airlines without interline arrangement in place. Another way to do it is, in the worst case, collect your baggage, take them to Jewel early check-in area (as long as you have at least 3 hours prior to departure of your BA flight) and drop them off there.

Q2 No visa requirement for UK nationals for Singapore anyway, and it is not a problem going to Jewel. It's attached to T1 so take a transit to T1 from whichever terminal you arrive at SIN (rather than taking the foot bridge to it because it's quite a long walk). Just make sure to get a landing card for Singapore and complete it.