I'm only about 1000 miles northwest of you.
But I only work on SU carbs when they are screwed to something not British.
The dashpot on the top of the carb has a damper screwed in the top. It needs oil in the damper to run properly, but not to run. The dashpot just bounces around. If memory serves, the mini carb is facing the firewall, so you can't just take the airbox off and look down the bore. The next simplest thing is to take out the three screws holding the dashpot on and lift it straight up. It may come as an assembly, or the piston may stay behind. Don't drop the piston, because they always land on the needle and bend it, then you say bad words for a while. Now- look at the needle in the piston- there is a step on the needle right up close to the piston, which should be roughly level with the bottom of the piston. Sometimes they start to fall out of the sleeve, or the needle retainer falls out of the piston. Next, take a look down at the jet, which is the hole in the middle that would accept the needle- it should be maybe 1/8 " or so down from the body of the carb. The height of the jet is what sets the basic mixture, and on a side bowl carb, is also the cold start enrichment- their is a lever mechanism on the bottom of the carb that pulls the jet down when you pull the choke cable. If its more than 1/4" down, the jet, or the mechanical choke linkage, or the cable may be pooched. Now, go get a fire extinguisher, cause if you dribble gas, it will probably hit some component made by Lucas, and we all know about Lucas and sparks- the only thing they build that doesn't produce sparks is their ignition system. Have somebody crank the engine over. If the float valve is leaking, it will push fuel up through the jet. That is wrong, and needs a new float valve, or the chunk of dirt blown out of it, or some such thing. If there is no fuel pushing up while cranking, it either has no fuel or the right amount. Take the top off the float chamber and see if there is gas there. Get a container, stick the fuel line or the whole float bowl top and crank it again- should give you a reasonable amount of fuel. If not, look at filters and pumps, etc. Also, make sure the float valve hasn't screwed itself out of the float bowl lid. Clean stuff up, make sure the piston moves smoothly in the dashpot bore, reassemble it and it will be close enough for government work. There is really no simpler carburetor on the planet. There are no holes in an SU small enough to get blocked by anything smaller than a rock, or perhaps a complete mouse cadaver. They always work, but never work really well. They will not start cold without some enrichment.
If it all appears good, I'd pull start it. With the lousy Lucas ignition, and the lousy Lucas wiring, and the lousy Lucas starter drawing too much power and cranking too slowly, it may just need a bit of abuse to go.
Sorry about the Lucas digs, but I just can't be completely polite anymore with Brit cars. Too many memories, too much pain...