Whilst attempting to claim against LOT is likely going to be the easier option, they will only be liable for the 2nd leg of your journey. LHR-WAW is 1,474km, which means it's considered a "short-haul" flight under EU261 legislation, and thus only eligible for the lowest level of compensation - a maximum of €250.
Whether BA is liable or not is a tricky situation - or perhaps it's not. UK261 (the UK's equivalent of EU261, which is basically an exact copy of the law that was put in place after the UK left the EU) states that the delay is considered based on the arrival time at your final destination. But is BA really liable for a delay on a 2nd flight on a different carrier? That's hard to say...
I think there's 2 possible scenarios here, based on when the flight that BA rebooked you onto was was DUE to arrive in Warsaw (ie, not including the 10 hour delay).
If that flight was due to arrive more than 4 hours after your initial scheduled arrival time into WAW, then BA is absolutely liable. The initial delay was theirs, and they rebooked you onto a new flight that was due to arrive more than 4 hours after your planned arrival. The fact that delay actually became 14+ hours isn't relevant - it's still more than 4 hours! Under UK261, they are liable for compensation of £520
If that flight was due to arrive less than 4 hours after your initial scheduled arrival time, then it becomes a more complex situations. Personally I would argue that they are responsible for triggering the entire delay, but I think you'll have a hard time getting them to agree in this situation. If it were me, I've probably just go after LOT in this case.