Based on the Sophia airport website, with its lack of information on left luggage options, combined with the fact that several other sites mention that there is not left luggage on the airport, I would say that you can't leave your luggage at the airport.
Wikitravel has a page about Sophia and they do mention luggage several times on the page, including several warnings about paying extra on some kinds of public transport.
If you are visiting Sofia on your way to other destinations, you can leave your luggage and tour the city for a few hours. The Central Railway Station is located right next to the Central Bus Station and offers luggage storage. It is some 10 minutes walking distance from the Lavov Most (Lions' Bridge) area and the city center. To find the Luggage department at the Railway Station look for old train placed as decoration in front of the central entrance. You can leave your luggage there for a period of maximum 5 days . Also a luggage Storage office is working in the Central Bus Station, for storing luggage for 24h the price is 2Lv(July 2014) per piece.
This quote is from Wikitravel, see link above, and from that I would say the maximum time you can leave your luggage is 5 days, which might be a bit short for your needs.
That likely leaves you the traditional option if you will have been a guest in a hotel in Sophia before you head off to England. Contact your host and ask if he is willing to hold your luggage for you and what he will charge.
There might be the option of 'self storage' but I do not expect that kind of places to have an English language option, a quick online search does show that their might be options.