It has already happened. The pre-1998 loan book has already been ‘sold off’ to the private sector.
The transaction is fairly simple. The bank gets the loan book in exchange for reserves, and the government then pays them the repayiment based upon what HMRC gets in payroll deductions.
It just become an off balance sheet index linked Gilt, but where the tax has been essentially privatised (HMRC ends up acting as agent for the private sector). Arguably payroll deductions for compulsory pension savings are in the same category. It’s just taxation paid to a private company to try and make it look like it isn’t taxation.