It's primarily the executor's responsiblity to pay the tax and the executor, of course, has or should have access to all the deceased's financial records, including the records needed to compute this tax liability.
Not sure what happens if the deceased has failed to keep proper records, but this is a specific example of a general problem that executors may face — the deceased's records prove not to be in order and so they have difficulty in identifying and paying the deceased's liablities, including outstanding tax liabilities. I suspect the answer is that the executor retains a tax adviser to assist them in dealing with the Revenue to agree the liability.
Tl;dr: Keep your records in good order, folks. You're doing your dependants no favours if you leave them a mess to sort out.