I've done similar on an all BA itinerary, twice. When I checked in, the agent asked me if I wanted my bag in London or if they should tag it through. I can't tell you how liberating it felt to exit LHR with just a backpack, hop on the Tube without lugging my bag, and easily go to my hotel (I picked a place almost across the street from a Tube station).
My advice would be to have a backpack or something small that you'll have with you that has your essentials (including medication, change of clothing, etc.) and check whatever you don't need for that 1 night.

It might depend how late your incoming arrives at LHR and what time you leave LHR the next day. The first time I arrived around 7pm, the second 3pm. Both outbound flights left around noon the next day.