Hi,
First a question: Which Webers do you have installed on the car?
If they are downdraft, I can't really give any suggestions.
If they are sidedraft such as DCOE, I might be able to help. On the chance these are what's on your car, here are some ideas.
I've had a pair of DCOE on my TR4 for 25 years and the chokes have never been operated (there's not even been a cable connected). The "choke" on a sidedraft Weber is really more like a small, separate carb or enrichment circuit, not really a choke. It's also pretty much worthless.
Since DCOE and similar have an accelerator pump, the method I use (and I've heard widely recommended) is to just forget about using the "choke", fully pump the pedal two or three times, then hold so that the throttle is slightly open. Crank the engine and it should kick over, but might run a little rough at first and not want to idle. Use the pedal to hold the throttle slightly open for a minute or so, until the engine settles down and wants to run and idle more smoothly.
If it doesn't start first try, repeat up to three times.
After that it's likely the engine is flooded and might be necessary to clear it by holding throttle wide open (no pumping... just slowly press the pedal to the floor) and turn the engine over, releasing the pedal partially as soon as the engine starts.
Hopefully this helps. DGV and other downdraft Webers are different animals and I honestly don't know how their chokes are set up or if any of these techniques will work.
If problems persist, I'd suggest checking timing, spark plugs, wiring, points, valve lash, fuel supply, fuel pump, carb balance, carb float settings, cleanliness of fuel/condition of any filters, etc.
p.s. Also, do you have an electric fuel pump? One would do a better job filling the float bowls at start-up, prior to trying to trying to start the car.
A Facet pump, for example, is small and can be hidden under the rear of the car or in the trunk if you wish, might be used in conjunction with or as a backup to the original fuel pump.
Using an electric fuel pump, exepcially with Webers, fuel pressure needs to be regulated to about 3 or 3.5 psi.