Could the UK have a Muslim prime minister?

From purely a question of law, yes, there is no religious bar to the office of Prime Minister of the UK.

From a question of societal pressures and how Britons would feel about a Muslim coming to power as a Prime Minister, there are likely to be more obstacles. There would be two main arguments: the first would be how a member from a minority group could be expected to represent the will and views of the majority culture and the second would be a question of ideological loyalties. The first argument is an attempt to mask the cultural issues brought up in the second.