Swastikas weren't just a Nazi symbol, you find them on Roman stuff (eg Lullingstone Roman villa, somewhere - presumably near Lullingstone , England)
There's photos of Brisfits over the middle east with them, pretty sure there are ww1 photos where the occasional rfc aircraft sports one. Never very often.
We crossed the idea of the swastika very early in my degree course, if I remember, it was an indian symbol of good luck (among other cultures) or something like that, so my educated guesstimate is that the british, being at the time kind of a big deal in India, adopted it, along with various other bits and bobs.. Food, words, symbols etc. That's where the british got it. It's a bit like people slapping the ace of spades everywhere in various militaries (US, I'm lookin' at you in particular) - just another interesting symbol although it rather went out of fasion when some arsehole from austria took a shine to it..