If you ask, they'll almost always say yes. Something like, "Hey, I noticed the flight is mostly empty. Would it be possible for me to get a complimentary upgrade to one of the empty first class seats?"
Unlikely. There are different crew for different cabins. When a flight departs without anyone in first, for example, they don't normally bump people up to fill it they redistribute crew into the main cabins.
Now, if you fly regularly and have some miles status with that airline you can request all sorts of things. The gate agent has a lot of leeway in that regard.
Had an empty flight with Singapore Airlines right after Christmas and they didn't.
This is British Airways so we'll see.