They think that, when they reassembled everything after replacing the timing belt and vacuum pump, a sensor that controls the amount of fuel delivered to the engine malfunctioned ("was left open"). This caused too much fuel to enter the cylinder, the fuel ignited prematurely and this caused the timing belt to skip.

I don't believe it. For one thing, I don't think the fuel injection system needs to be touched during a timing belt change. For another, if 'too much fuel entered a cylinder' it would merely incompletely combust and emit clouds of black smoke. And another thing, it would be impossible to make a timing belt to 'skip' a few teeth unless the tensioner had been reassembled incorrectly and wasn't tensioning, which I suspect was the cause, unless they got the timing completely wrong - also possible.

They clearly mucked up the job and are trying to think of the most inventive and superficially plausible excuse possible. I would complain directly to Peugeot and insist on a new engine, as if they have had to replace valves and camshaft, the engine has had a huge shock and might possibly be weakened elsewhere. Do keep us posted....

Agreed.

I was an amateur mechanic. What they are describing is impossible

They screwed it up.

I have replaced a number of cam belts and the motto is check , turn the engine over and check again. Never had a problem. And as long as they did that , it would have been OK. Fuel Injection does not go wrong arbitrarily. They are lying.

Update: Requested Peugeot to either replace the engine or buy back the car, Peugeot refused. Car was supposed to be handed back to me on Friday just gone, but developed another fault so is still with the garage. Waiting to hear more details of the fault and have told Peugeot I will respond to their refusal to replace the engine when I know more.