Austin Princess - stumble setting off, poor starting

First thing that comes to mind is that fuel is draining back to the tank, and you're not getting a good supply? Or the fuel pump has a small hole in a diaphragm and isntsupplying enough for the engine when it's cold?

Edit- I'm not that well up on SU carbs, is that the one with the float chamber on the side, or integral?

Typical, I'm all out of monkeys! 🤔

Edit- with apologies to vulgalour for the random direction his serious thread has taken!

As above, sounds like it's not getting full choke, I'd check the cable and linkages first then perhaps the float level and the piston isn't sticking.

Fuel pump diaphragm - unlikely. It's on an in-tank electric rather than an external mechanical. The pump is also only a few years old.

SU Carb - integral float rather than on one side. Carb was rebuilt fully not too long ago with new parts and is quite clean, the float level appears to be correct, I'm not getting any flooding issues and, as far as I can tell, no bent needle, sticking float, or worn components. There was a suspicion the throttle spindle might be at fault but that seems less likely now. No poppet valve on the flap either, just a solid flap on this carb,so that's not going to be the issue either. I did also check for vacuum leaks, particularly around the spacer where it bolts to the manifold, and there doesn't appear to be one. Car sat about for a few months only so I wouldn't expect the internals to be that bad, but I may drop the bottom plate off to see if there's any sediment in it. This carb hasn't been particularly prone to getting dirty inside from lack of use.

Choke - it is indeed a cable type on this. I am getting full motion of the choke cable both at the carb and at the pull inside the car. I'm not getting any restriction or binding, and there's no excessive slack. The only issue with the cable is that I have to use the traditional clothes pegs to keep it in place since the locking mechanism doesn't lock. The pegs aren't slipping out of place or allowing the cable to slip.

If I can get it do to either of these things its doing on camera I will, it might make diagnosis easier. I suppose it could just be Old Car Stuff, but I've had this car running without these problems for a good amount of time so that's what makes me think something is amiss.

I take it you've got enough oil in the carb dashpot? My Traveller with HIF4 is very sensitive to this, and gets hesitant on acceleration if it's running too low.

Actually the dashpot is a good call for the starting issue, I thought about when I last checked it and it was... before taking the head off*. So, it's entirely likely that's a bit low. It would also explain the intermittent nature of the stumbling issue because I've never checked that against dashpot levels so may never have just linked the two together.