In fact I think I can think of a counter example. Internet access has taken longer to grow in Germany than the UK for example, despite Germany being more prosperous. Yet JW growth has been more sluggish in Germany over the past two decades than in the UK. It's weak evidence either way, yet if the Internet significantly slowed growth we might expect the opposite pattern, ceteris paribus.
Actually I believe this is probably a good example of your previous point . Until *very recently* I'd be willing to bet there is a lot more net migration into Britain than Germany and with our colonial past a good deal of these people were from 3rd world Commonwealth countries, such as the Caribbean and Africa, with a traditionally very religious culture. I believe that's were you'd find the growth, rarther than from native Britons. Unless the birth rate of the two countries is significantly different.