Vish the problem is demographics given what we know and experience it's obvious Muslim and none Muslim populations don't mix there are exceptions but that is the experience across the world. So to go to a position where Muslims are the majority in Britain France Germany etc would be national suicide. And you can't deal with integration when this situation prevails.
But your immigration is never going to be more than your locals and you will control that. I think you have less than 5m Muslims which is less than 10% of you population. I don't think that all migrants will come from Muslim countries or you have to accept all applications. Yes, you could have lots of migrants from the EU before the Brexit, but, the Muslim population in the EU, especially Eastern Europe is low. With your ageing population you will need to increase migrants, there is no choice if you want to grow, do you think it's wiser to open doors to Poland or Pakistan? Whose citizens are more likely to integrate? This is why Turkey was never accepted in the EU
To be fair most of the immigrants to Britain are Christian but certainly France and Germany need to take care merkels policy was lunacy for which Germany is now paying. I don't get the ageing population argument with the new robots coming into the workplace demand for labour will be lower unskilled labour almost nil any employment will require skills in the future. Robots soon will even drive vehicles.
I read that the Muslim population in UK is a bit less than 5%, about same in Germany, while in France it is a bit less than 10%. The bigger share of those comes from the colonial past for France and UK while in Germany it is the Turks who supported Germany in the world wars. France had more Muslim colones then UK, this is why it's like that. The recent wave of immigration didnt add a lot except in Germany.
Regarding ageing population, every economist and actuary talk about that, but I know, you don't believe in experts. You think robots will replace nurses and hotel employees very soon it looks, this is not what I see happening at least that soon. Even picking fruits need hands and can't see robots replacing them so soon.