has given you good information.
I the ignition and fuel pumps are on fuse 12 in the original Shadow series cars (see PDF pg 5, Chapter M, Workshop manual
One relatively easy way to determine if you have spark, but not fuel, is to detach your air intake line from the "center tube" that joins the section between it and the carburetors. Give a light spray or two of starting fluid then try to start the car. If it sputters or, like mine did, runs perfectly for a couple of seconds then dies you know you've got spark but no fuel.
Check that you've got power on both sides of the fuse using a multimeter. Touch the positive lead to one side of the fuse and the other to ground, then do the same on the other side of the fuse. You should have 12V on both sides (or mighty close to 12V).
If that checks out, go on to checking the power at each of the pumps (the technique is the same, positive lead to power input and negative lead to ground). Again, you should have something around 12V. You will need to slide the rubber end caps off to get to the power connection at each end. If the wiring is original Crewe used wire that was white with a pink stripe.
These cars can (and often do) run with only one of the two dual pumps operating. You won't have a "failure to proceed" unless both pumps fail. I think my car was doing that for years. If you're really ambitious, you can install a discreetly hidden SPDT switch that will allow you to isolate one pump from the other to test them individually, but that's another story.
It is entirely possible, and has been done many times, to splice a more typical disposable style gas filter into the line. I would not operate the car without this and both the carb filters in place. By the way, the Crosland 444 can be obtained from HVAC supply shops in the UK, as it is also used as the filter for oil furnaces. It's just a pain to locate a supplier who still has stock. There are also aftermarket options to replace the canister filter that's on the car that still have easy replacement part availability. The one thing you don't
want to do is to throw away any of the Crewe Original parts you might remove. Who knows what future owner might want to "go back to original" and not having these parts can often kill a sale.
Rebuild kits for SU carbs and fuel pumps can be sourced from Burlen in the UK. Also Victoria British
here in the US can source rebuild kits for the fuel pump. Our cars use precisely the same kit as an MGB (although I would reuse the diaphragm spring that is already in the car if there's any doubt about the spring pressure being the same). The part number from Victoria for a pump rebuild kit is 3-2058 (and you'd need two). As of last week a new pump from Burlen (an SU AZX 1405 series pump) was about $265 with shipping. Two rebuild kits from Victoria British are a bit under $150 with shipping.