"set" in this context means everything shows that something is going to happen. I-e the tendency is so strong, so one can accurately predict the outcome. If you are "set to succeed" for example, you have planned everything and worked hard and it's almost certain that you will succeed.

'set' here means prepared. 'set' is used to get athletes to get ready before a race. You set your alarm clock. Using it in this context is actually pretty meaningless. It adds nothing to the sentence. Perhaps the journalist had space to fill up?

I think the meaning of this sentence is: In the UK, the population growth will most likely remain flat. That means that probably, the UK population will neither rise nor decrease. So, to answer your question, I think that ''set'' in this context expresses the likelihood, the probability of the population remaining stable.