Having done some further investigation, I agree with you, Warrior, that the youtube video "Muslim Demographics" contains a fair amount of alarmism, some distortion or referenced material, and a number of false claims.
I think this source gives a much more balanced picture.
Do Muslims Have More Children Than Other Women in Western Europe? - Population Reference Bureau
In particular, it documents that the birth rates of Muslim women tend to fall over time, as they acclimate to the culture of their new countries. However, it does also document that Muslims in European countries, including the ones who are 2nd and 3rd generation, and not only the immigrants, have approx. one more child per woman than non-Muslim European women.
So, while that video was apparently put out by an alarmist Christian group that would like you to have more non-Muslim kids, I do believe that there is a very large kernel of truth that they have built upon.
In particular, that is that 1)Muslim women in EU countries have a much higher birth rate than non-Muslims, 2)non-Muslim populations in EU countries are declining, while the Muslim population is growing, 3)In Europe, women with strong religious beliefs have more children than women who don't have strong religious beliefs, and 4)Muslim immigration to EU countries continues to be a factor in the demographics of EU countries.
With families having fewer children, and mostly wanting them to get good educations, and find professional employment, which is a trend I also see in the US, then immigrants come to fill the jobs of laborers, and workers in manual jobs. That is a trend that shows no sign of abating, as far as I can see. In the US, that means more Latinos, and in the EU, it means more Muslims.
So, whereas, the time periods trumpeted in that video seem to be distortions, based on playing fast and loose with known facts, the overall trend they are pointing to, does seem to be correct.