Everything ksand said is 100% true.
However, it *is* likely you will be okay as you don't have an application processing.
Your toughest challenge will be boarding the flight, as I always had to show my visa as I didn't have a return ticket. The airlines are charged if you are denied entry and sent back, so they try to eliminate that risk.
At the border, especially as you have a copy of your BRP, it shouldn't be too much trouble. Worst case will you be have to sit in the pen while they check all your details, likely give you a telling off and tell you not to travel without your BRP again, and send you on your way.
If you had an application in with UKVI, it'd be a non-started. But as it's with DVLA, I'm hopeful they will have a bit of compassion for you.