That pretty much takes the AC servos out of the equation, as 3 of them will put you over your budget.My budget is max around £1000.
I don't have any experience with the closed loop steppers, but I can tell you that I've never seen a DIY router with Nema 42 steppers. You definitely don't need motors that big.
I don't know anything about your machine, but some mid sized Nema 34's, and good digital drives (standard open loop) should be more than adequate, and a bit cheaper.
UCCNC is an open loop control. If you want full closed loop, you need to go to LinuxCNC.
If you stay with UCCNC, get the UC300ETH, not the USB version.