tyre pressure monitoring sensor keeps flashing up, saying it can’t detect tyre pressure. Could this be a faulty tyre sensor? 

Yes.

Did 2001 Mondeo's have TPMS ? I'd have thought not.

Or are we talking about an entirely different vehicle ?

Unfortunately the battery can't be changed, you have to change the whole unit.

For a back street garage you're looking around £75 per wheel. 

1 hour ago, Bomb Jack said:

For a back street garage you're looking around £75 per wheel. 

I'm pleased I don't live anywhere near Cheshire. 😉

Set of 4 new TPMS sensors can be bought for £25 and here in the North East the 'back street' garages charge between £15 and £20 per wheel to fit them. Total cost for 4 wheels including the sensors is £105 maximum.

Google says the following.

Replacing a TPMS sensor in the UK can cost anywhere from £40 to over £100 per sensor, with total jobs ranging from around £20 for simple fixes to £200+ for all four, depending on the car, sensor type, and if it's just the valve or the whole unit, plus labour for fitting and programming. Expect £20-£55 per wheel for replacement/service, plus potential costs for tyre removal/refitting, and it's cheaper to get a new sensor cloned and fitted than buying OEM parts from a dealer. 

I decided at the time to use the mechanic that did my MOT (the first one not done by Ford in my car's history as they were closed for a period).

He's obviously not as cheap as up your way unofix, lesson learnt for next time.