How long 198KHz keeps broadcasting is primarily dependent on how long the remaining valves last. There's a letter from the BBC that says they will get to at least 2016 when the BBC's charter is due for renewal, at which time the Government will need to decide if they really need the transmitters to be replaced.
https://www.elexon.co.uk/wp-content/...ram-180712.pdf
Valves do have better protection against electromagnetic pulse damage in the event of a nuclear blast, but when they're connected to a massive aerial array that probably doesn't help. It might be easier to restart after though unlike VM's network which would almost certainly be toast.
There's a good piece on the history of the Droitwich transmitters here:
http://www.bbceng.info/Operations/tr...ch_calling.htm