I don't see A magic arm race kicking in until at least half the world is effected by the return of magic. It would be to no ones benefit to do anything like that if it could only be used In england or some such place, but that you would lose the power to cast spells when you were in a magic dead zone. (if i am remembering correctly).
On top of this, Its hard to train wizards in something that you have very little knowledge to train them with. Magic has been dead for a very long time and a lot of knowledge was lost. On top of this- none magical folk are going to be distrustful after the first bit of wonderment has passed. Old superstitions will come back in to popularity. check out Shadowrun's timeline- they covered this pretty well; it was mass panic in a lot of places.