Does either the UK Government or Prime Minister have any kind of vote or power in order to remove a particular EU Commissioner?
No. The government of each member state nominates a member for the Commission, but cannot later give them any orders or remove them from their post. The members of the Commission are to act in the interest of the European Union as a whole, not their home country.
The only ways for a member of the Commission to lose their post before the end of the Commission’s five-year term are (according to TFEU, article 246) if they die, resign, or are compulsorily retired. Compulsory retirement happens when the Court of Justice decides (on application by the Commission or the Council) that a member of the Commission is either unfit for office or is guilty of “serious misconduct”. The President of the Commission can force any member to resign, and the European Parliament can force the whole Commission to resign by a vote of no confidence. That’s it; individual member states have no say in this.
(As a special case, the High Representative of the Union for Foreign Affairs and Security Policy – who, by virtue of their office, acts as a member of the European Commission, too – can be removed from office by the European Council. The European Council also appoints the High Representative. But again, not by any individual member state.)
That's only possible in case of misconduct or the like. Concretely, the UK has a vote in the EU Council (whose purpose it is to represent the member states' governments). They can therefore vote to remove a Commissioner by referring him or her to the EU Court of Justice, which can be done with a simple majority in the Council.
[A simple majority means 15 countries, no matter how large or small. The UK, Germany, France or Italy therefore do not formally have any special weight on such a decision. Most Council decisions are not taken by simple majority but through a qualified majority of 55% of the member states comprising 65% of the EU population.]