Everyone is responsible for ensuring they pay the correct amount of tax regardless of whether they're employed or self employed. Claiming ignorance is not going to wash i'm afraid, everyone knows they need to pay tax. this extract is taken from the CWG2 book, page 17, which can be downloaded.
12 Deliberate deduction failures
In certain circumstances where we consider an
employee has received pay knowing that the
employer has deliberately failed to deduct tax from
the pay, a direction can be made for the employee
to pay the underpayment. Where we also believe
that an employee received pay knowing that the
primary contributions had not been deducted or
paid over, a decision can be made for the employee
to pay those contributions.
Primary contributions are BIL's NI.
However, it's also the uncle's legal responsibily as an employer to calculate and deduct tax and ni so if the decision is made that the negligance is on his part then he will be liable for some, perhaps all, of employer and employee contributions. It will really come down to HMRC to look at the case and decide who will foot the bill to be honest. The best thing for your BIL to do is phone the local employer compliance department and come clean, it may eventually work in his favour if he does.