One possible reason is that many of the people with whom Islam is growing rapidly tend to from areas either in conflict or in extreme poverty, usually both. As cited by the Pew article: two areas of high Islam growth are the middle east and sub Saharan Africa, which have both continued conflict, and extreme poverty. While the ME has oil revenues, precious little of that money filters down to the average person.
In desperate times, people are more inclined to turn to religion for answers, because their current society isn't functioning. And in both the middle east and parts of Africa, Islam is the established religion. Religion offers a form of stability where none exists. One would have to live under conditions of poverty and conflict to understand how powerful a motivation that can be.
The desperate situation also accounts for much of the higher birth rate... when life is cheap, as it is with both conflict and poverty, some of the first to suffer and die are the children. The birth rate tends to go up in the hopes that some of the children might survive, and tends to stay high for a while even when the threat lessens.
In Europe, which has seen a huge influx of Islamic refugees, the rate might appear to be growing, when in fact it is just people relocating, courtesy of a nearby conflict, an overland route, and an immigration policy crafted before the volume went sky high. Not more converts, just more people adhering to Islam appearing where they didn't exist in large numbers before. In the absence of context, this might leave the average European prone to thinking that Islam is growing at an incredible rate, when in fact it is growing at a fairly slow rate.
No surprise that British publications would jump on that Pew report... it's the sort of alarming prediction that sells a lot of ads.
As for the claim that Islam will be the world's largest religion, that assumes that it's current growth will continue at a linear or accelerated rate.
Any number of things could change that, from further wars in Africa that result in a high number of deaths (Rwanda on a larger scale is quite possible), to many of those people achieving stability and not needing a religion to lean on, to the rising wave of women's rights coming into conflict with the somewhat dated view of women that much of Islam takes.
The Roman Catholic Church was the fastest growing religion at one point, until Martin Luther nailed a few comments to a door, and Henry VIII decided he didn't like his current wife.
A lot of things could change to slow or reverse that trend.
Finally, remember that Islam isn't one religion, any more than Christianity is one religion. Islam is several religions, that don't necessarily agree with each other. Interesting that the Pew report doesn't break it down into Sunni, Shia, Alawite, Sufi...