This fall right into my area ... in our resto/custom projects part of the deep cleaning and "renewal" of interior parts includes heavy washing to get rid of several years of dirt, and the foam soaking is a classic occurrence.
Sorry to be blunt and drastic, but with the proposed ideas will be nearly impossible to get rid of the wetness, you might also incur in a mushy smell that you might never get rid of, or the foam rotting and disintegrating, which I seem to understand is already happening.
The type of cells of the foam are not made to hold water, nor they easily release it ... I've had a car leaking water in (it is a parts car in our wrecker yard) and even after a full summer with little rain the bottom part would still be wet ... I've had another in which I was unable to pull the carpet out to access some connectors, the water turned to ice (it was winter) and it "glued" the foam to the metal :-)
The P38 has a very very thick foam, even by other brands' luxury vehicles standards ... really high and heat formed very precisely to the metal floor. Don't cut it, even if partially missing it still keeps the shape somehow.
The only real solution is to take it out and let it dry, the way Clive did.
Would propping the carpet up and using a heatgun on low allow the foam to dry out do ya'll think?
Propping a piece up and putting heat is a miserable solution with questionable results. DO NOT use a heat gun because the concentrated heat melts the foam, which contracts sharply and makes a horrible smell (don't ask me how I know). At most, you can use one of these small heater like for the bathroom (obviously you need a mains... uh, you guys in UK got what, 110V?). And you need to think to leave it for hours and hours, and you will never be able to get all the foam to dry anyway, since whatever remains in contact with the floor is still ... wet. And besides the time and electrical expense and the risk of melting down the heater directly, you also need to keep the car ventilated from time to time .... ah! and you might end up twisting/deforming the carpet - depending on the "propping element", which will probably ruin it. Been there, done that :-( (all of it including the heater meltdown lol)
What Clive suggests, to do we do it regularly in our workshop with the vacuum cleaner/carpet washer (we use a Karcher SE400, borderline between a home appliance and a pro machine, but okay for regular use).
But only when you cannot get the carpet out, otherwise the best way is to really take everything out, power-wash it with good household cleaning materials, and then use the vacuum to take the biggest drip and then let it dry naturally. Obviously is a nice season job.
This type of "wet vacuum" machines can help you take some of the water in the foam underside the carpet, but it is a long an tedious job. Nevertheless, if you cannot get the carpet out now, and you have access to one of those, you can give it a go for the time being and guaranteed will remove some of the water.
Sorry for the long post. Now, some candy ... here's the underside of the carpet, see how precisely it fits!
And here is being washed .... lots of dirt, it changed actually color