Does it matter in which country I obtain my EASA Part 66 AME licence? I have completed EASA Part 66 B1.1 exam modules (some from the UK CAA and some from the German 147 organisation).
My queries are:

Can I apply to any member state (e.g., Belgium, The Netherlands, Germany, Ireland, UK, etc.) to get my initial Part 66 licence?
I have already emailed few country’s licensing departments; they declined and advised me to apply only to the UK CAA.

If I can apply to any member state, why did I get these replies? If I can’t, why not?

Does it affect job prospects in the UK if someone has a Part 66 license issued by another European country? (And vice versa?)

To which authority (country) can I apply to get my Part 66 licence as soon as possible if I am eligible to do so.

Does it affect aircraft type rating endorsement in future on a Part 66 licence? For example, if my license is issued from Belgium but I’m working in the UK.
When you have to deal with the authority, the theory and the practice varies a lot. So, there are things that theoretically you should be easily able to do, but that the authority won't accept, for no real reason. Sometimes it's just how the guy at the desk woke up in the morning. 

Anyway:

1) Yes, you can apply in any country of the EU28. I know Switzerland won't accept your appliances unless you work in Switzerland, don't know about Norway and Iceland. Have no idea regarding the new situation in UK after the Brexit. 

2) Because Europe is not a federation of States like in USA, and each authority want to impose own power and rules and protect our market. Ask them why.

3) Theoretically no. Pratically, maybe. 

4) All EASA countries. Where do you work? Where did you get your theoretical modules and practical experience? Try starting from there.

5) Repeat the question again, it's not very clear to me. 

To apply you need to compile and send the EASA Form 19 (You can find it here) to any European competent authority. 

You can also find the Annex III, the Part 66 of the Easa Regulation 2042/2003 here, with all the pertinent information. (In particular check the 66.A.10 Application)