Check out the conditions for each airline.
Direct from UK: Check the luggage allowance before you fly with each airline that operates, they may differ.
Via another country: if travelling via Italy and using a Neos plane for example, they only permit 15kgs of stored luggage!
If you are over your allowance it may cost you £10 / 10euros per kg extra.
On direct flights you may want to consider booking an upgrade, it will cost more but there are a number of perks including a higher luggage allowance.
I am not sure about the legalities of leaving your baggage in left luggage, as normally when leaving UK we are asked who packed your bag, whether you have left it anywhere etc, so leaving it in another airport may/may not be questionable.