We've lived in Wembley for about 5 years now.
I don't love it, but there are some plus points.
Transport links into central London are really good, if a bit long (Bakerloo and Euston overground from Wembley Central plus a few mainline trains, Jubilee and Metropolitan line from Wembley Park plus mainline to Marylebone from Wembley Stadium station).
There's so much regeneration going on, both around the stadium complex and along the high road, that I'm sure the area will change over time, but it's not going to happen overnight. There's a new(ish) French school over by Wembley Park which is attracting French families to the area, changing the demographic a bit.
Big south Asian community. If you like south Asian food there are loads of great little cafes along Wembley High Rd where you can get a tasty lunch for £3-4. Ealing Rd is known as little India - lots of colourful sari shops, grocers etc. Hindu temple.
Bit surprised to hear PPs talking about crime as I find it generally feels quite safe, lots of families and not much of drinking culture (except for sports fans visiting on match days). My general impression is that there are some other more central areas where crime is more of a problem - Kilburn, Harlesden etc. But I haven't actually looked at the figures so am probably wrong. We've never had any problems.
We moved here for similar reasons to you - more space / garden compared to our previous more central location, good transport links, convenient for mine and DH's work. I generally find it a bit frustrating that (apart from our fave Indian cafe, and the local Boots/Tesco/Wilko) I don't really use any of the local amenities, so find myself leaving the area quite a bit. Our bit of high street is mainly betting shops, "sell your gold here" type jewellers, pound shops etc. But house prices have meant we've been able to afford a little 2 bed terrace with garden which we LOVE, in a little cul de sac with lovely neighbours (mainly elderly people who've lived there decades).
Sorry that's turned into a bit of an essay! In short I probably wouldn't recommend it, but it does have its upsides. Have you come to have a look around? I imagine it'll be quite a contrast to lovely Cambridge!