I imagine that old Satchwell stat is switching the electric UFH mat directly. I believe its switch contacts are rated at 16A, not that your mat is necessarily taking the full 16A!
I believe many modern UFH timer stats are designed for wet systems, where they switch the boiler or a valve with much lower current. So you would need to get one suitable for the load of your floor.
This sort of thing would do:
www.heatmat.co.uk
I've used one of these - good but I had one fail!
Warmup’s range of programmable thermostats for underfloor heating, including digital and automatic heating controls. Offering you big savings on your energy bills.
www.warmup.co.uk
There must be others available with 16A contacts!
PS I'm assuming your black wires are actually neutral, I guess that's pretty likely, but worth checking if you have a meter.