perhaps flights from a London airport (Gatwick or Stansted) will fly direct to Varna.
Both BA (Mondays/Wednesdays/Fridays/Saturdays) and Bulgaria Air have regular scheduled flights from Gatwick to Varna (Mondays/Wednesdays/Fridays) - the major issue with flights concerns regional airports, where tour operator flights to Varna have been withdrawn in favour of Burgas.
You can find details of Albena HERE. I'm no lover of the large resorts in general but must say that Albena is the best of the larger ones that I have visited in Bulgaria in terms of being well-maintained and managed (apart from the management company's decision to go predominantly 'All-Inclusive').
There are many resorts, both large and small, from Krapets & Shabla (near the Romanian border) Southwards.
