Yes, they do send your old passport back with the new one - they just cut the corner off to show it is no longer valid (I have my valid US visa in my old passport so I carry 2 passports with me when I travel to the US - my old, expired one with the visa, and my current, valid one)
You should maintain a valid passport in order to show your UK citizenship for the FLR(M) visa (otherwise you have to send your birth certificate and several other documents to prove your citizenship instead), so I would just go ahead and renew your UK passport now so that you continue to hold a valid passport.
If UKVI query the expiry of the passport you provided, you can send them a copy of the new one when you get it. Looks like it's taking up to 4 weeks at the moment to get a new passport if you apply online.