Does the BBC still have these recordings and what are the chances of getting access?
I'm not sure what the overall situation is. I know that quite a few have been obtained for commercial release. For example, a good selection on Bon Copper's collecting in Southern Counties was released in 1977 on Songs and Southern Breezes Country Singers from Hampshire and Sussex
Various Artists Topic Records 12T317 (LP, UK,) and much later a lot of the Voice of the People recordings by Kennedy and by Copper were originally from those BBC collecting days.
As Marie Slocome writes in the article that I linked to above, these was originally transferred from tape to 78rpm shellac in the 1950s and then later to 33 1/3 rrm vinyl LPs when that became the standard. However, at this stage a process of selection was made and not everything was transferred to the newer format. I know that this grieved Bob Copper because some of his favourites that he had recorded were not among those that were selected for transfer.