As far as the computer side of things goes, what is your computer experience and can you beg any help from family or friends with better skills?
Mach3 will set you back about £150, a second hand PC from ebay etc £50-£100. This eats into your budget a fair amount.

Alternatively you could look at the Machinekit OS running on a beaglebone black (£40) with a PBX-BB bob (£60)
The BBB has a huge advantage because the processor is specifically designed for industrial motion control with 2x 200mhz PRU's that deliver much better performance than a windows x86 PC could ever do.

It will save you £150-£300 on computer/software/bob, as long as you can manage not to runaway screaming from the linux.

For stepper motor drivers, the TB6600 drivers can be purchased for £9-£10 a piece from aliexpress, buy them one at a time and avoid going over the £15 import duty limit.

You can build a much, much better machine for your budget, so don't be disheartened.