2001 Citroen Xsara Picasso trouble starting the engine when is cold out

1st thing is your mechanic isn't up to speed on Hdi engines,
the heater plugs aren't really in use for cold starts on these engines.
The cold start function doesn't come on unless it's well below 20 degree under 0!!
As you discovered they don't help even when you wired them direct.
So conditions that have to be met for a good starting Hdi...
Cranking speed has to be over 220 rpm,
Less than that and it does as you describe,
Good compressions,
and plenty of air.
Easy check is the air filter renew it they're cheap enough.
Next is the battery,
get it load tested,
this is with a dedicated LOAD test device NOT a multi meter!!!
At this point I would test the amperage draw on the starter motor.
I'll repeat this in big shouty print.
THE ENGINE MUST TURN OVER AT CRANK OFF THE KEY AT 200 RPM minimum.
If it's say 180 rpm or less then it will sometimes start if it's warm but as a diesel compresses the air intake charge to over 500 degrees so the injected fuel detonates if it cranks slow the air charge just isn't hot enough for that to happen.
Continued cranking does heat up the cylinder combustion chamber so a low speed cranking engine will eventually start like yours .

Make sure the oil is the correct grade if it's to thick it will struggle to start in cold weather.

Take 1/4 drives good , experienced advice .

I think the first thing I would check is the fuel pressure as its being a pain to start when cold.Aswell as a minimum cranking speed of 200 rpm you need 220 -250 bar of fuel pressure.Depending on the system fitted you need 2.5 bar of fuel pressure at the filter provided by the low pressure pump in the tank and a return pressure of .7 bar(Bosch system)
The siemens system is different as it doesn't have a low pressure pump in the tank.I would get your fuel filter replaced and make sure you use a good quality fuel and when it is really cold also use a diesel additive.
Hope we have helped.