This doesn't make sense as minimum wage hasn't gone up in the three months to November. It goes up every April. Possibly private sector bosses are pre-empting the pay rise they'll have to give in April 2023, whereas most public sector pay is subject to annual pay awards. But equally, private sector employers are free to raise pay again in April to remain competitive, or give cost of living payments or bonuses, which public sector employers generally are not, so they are hard to compare.
However, it isn't true to say that few people in the public sector are on the NMW, there are so many essential jobs where people are paid the absolute legal minimum in both the private and public sector.
Just looking at the NHS, as that's where I work. When NMW goes up next April to 10.42 every band 1 and 2 member of staff will also fall below minimum wage so will need to receive a raise. There tens of thousands of jobs in band 2, people who help keep the NHS going, like health care assistants, receptionists, technicians, lots of administrative staff, call handlers, clinic coordinators, support workers, housekeepers, domestic staff, ward clerks, porters.
Looking at the civil service jobs search, there are 20 pages of jobs advertised at minimum wage or thereabouts (18,525 for 37.5 hours/£19,760 for 40 hours). Local government and schools also have plenty of staff paid NMW.
However part of the problem with public sector pay is that trained professionals are not receiving pay commensurate with the level of responsibility, and stress, they are under. Plus the student loan and potentially postgraduate loans they have to pay back and high pension contributions further degrade take home salary.
For example, The first band of teacher pay is £25k-28k so between £5,240 (£100 a week) and £8,240 (£158 a week) more than a 40 hour minimum wage job. I've never been a teacher, but it seems like a stressful job. Nurses on band 4 and bottom of band 5 are on similar salaries. The job I do is non-clinical and pay in the private sector is over double for approximately the same job. The linkedin offers become more tempting all the time, but like many in the NHS I believe in the ideology of it's existence and don't want to contribute to it's downfall by bailing right now. This is wearing thin though.
Retention and morale is terrible in the public sector at the moment. Everyone is constantly being pushed for more with less pay and less resources while traditional benefits such as pensions are being degraded. People are of course free to leave to go to the private sector, but someone needs to teach children and look after the sick and these are trained professional people who are near impossible to replace. I don't know why some people are not more invested in ensuring the public sector works. It is for the benefit of all of us.