I do sometimes use the waterproof cover Rob but mainly when really bad weather is forecast. I also have the smaller bonnet and hood cover which I plan to use more often now I've found this issue.
I've been on the waiting list for a garage for over 6 months

Hi Steve
In future I would use your hood cover that covers the louvres on the top of the bonnet to prevent water ingress to the heater box. Ive never had the need to check my spigot end but I may have a poke down it now I know what it is!
In the meantime I would dry all the carpet inside. I have a Vax wet and dry vacuum cleaner but I would consider removing the carpet to the sides of the footwells at least on the worst side to see if you can pinpoint the water entry point. The carpets are glued and probably tacked in place too so if you think you could see the water entry point without removing the carpets then that would be the way to go.
As has been suggested I would check the bulkhead to body seal and wiper spindles etc and any other possible place where water might possibly get in including the holes that secure the footwell locker.
Looking at the black colour of your floor, it looks to me like this was a pre existing fault when you bought the car so a trip to the dealers may be in order when we are allowed even if your car is out of dealer warranty.
Another suggestion would be to get a local Morgan enthusiast to spray his hosepipe on the car whilst you crouch in the footwell upside down looking for leaks
On second thoughts however whilst I know you have some Morgan enthusiasts local to you, I wouldn't trust either of them with a hosepipe especially whilst crouched in a vulnerable position

Rob