I spoke to my local butcher back when I lived in Kingston and the culled deer goes to whoever is prepared to buy the whole order – so small local butchers don’t really get a look in AFAIK.
They usually over about 200 deer each year over the two culling periods, depending on various factors* which is too much for most butchers to take on.
There’s sometimes a protest at the start of the cull but I haven’t seen one since I started commuting via Ham Gate. Mmmm, ham.
*Some years it’s been as low as 78, others as high as 261.