H4U (Onur) are the only airline flying direct from the UK to Bodrum over the winter months.
If you have to fly into Bodrum, the only way you could do it is to fly into Istanbul from the Uk & get a short hopper flight down to Bodrum.
BA & Turkish airlines fly direct to Istanbul.
for the hops down to Bodrum try Turkish airlines, Onur air, Pegasus & Atlas jet
(Im not sure which of these 4 companies fly into Bodrum tho .. they all fly into Izmir)