Couldn't you bootstrap it, get £5k on a credit card for the taxes to exercise some of the options, use the profit to pay the taxes on the rest (or a further bootstrap)?
If the shares are not liquid 'cus the company is still private, then no you can't.
But the Revenue consider you to have received value in that "piece of paper" that you can't liquidate? That just seems like perverse tax law - why is it that way?
Surely if they're private shares that can't be sold the extrinsic value is zero, the private share value for tax purposes is no greater than the value of the option?
I feel like at least a phone call to a bank would be in order at that point. If it's that simple, surely some sort of mutually agreeable loan could be worked out.