Wonder if HMRC will pursue Uber for PAYE.
Undoubtedly. Currently, they are cracking down on personal service companies in the public sector, but all businesses are equally likely to be investigated if there is suspicion.
The HMRC approach is unequivocal - if they believe that an individual should be an employee, not self-employed, then they will start tax fraud proceedings against the employer - seeking to recoup all tax and NI that the employer should have paid (including employee contributions) and issue a penalty for incorrect tax administration.
The fact that the "employees" may have already paid tax and national insurance on their earnings is irrelevant to HMRC - the employer is deemed liable and, unlike in a court of law, the burden of proof lies with the employer to argue the HMRC is wrong in its assessment.
It's causing a lot of headaches in the charitable and public sectors at the moment - a change in law was hinted at by Theresa May at the Party Conference in her speech about corporations avoiding tax.