Is after 10pm unsociable hours?

Question

and does the rate of pay have to increase if at the minimum wage?

Answer 1

There is no requirement for the minimum wage to change for "unsocial hours".  Some employers, especially if influenced by a union, do p0ay a higher hourly rate for work done during the night, but it is up to them to decide what time becomes unsocial.  In fact the modern tendency, especially in the public sector, is to have a single rate of pay for a job regardless of the timing of the shift being worked