Specialist music on BBC local radio has been under threat, and declining, for years. You're lucky to still have such programmes on BBC Lancashire, Merseyside, Leeds, Shropshire, Gloucestershire ... Here in Hertfordshire I used to listen to folk programmes on Radio Bedfordshire, Radio Oxford, and Radio London. These stopped a decade or more ago and now there are now no local ones that I know of. At least, with the advent of the BBC iPlayer, I can now listen to these remote 'local' programmes.

I don't know in what way these programmes can be said to be 'financially viable'. Presumably they exist where their local audience is big and vociferous enough, or the presenter has a personal following (which has its own risk). They were threatened a few years back by the 'Delivering Quality First' programme - i.e. budget cuts and station mergers - and I'm sure that will happen again. I assume they have a (diminishing) budget, and a remit based on playing and interviewing local or visiting artists.

The background to these changes, as Guest said, is the alleged use by presenters of personal service companies to avoid tax (link). It's not clear to me how this threatens the future of these programmes more than they already are.