Fuel/air mixture problems
1. Could it be possible that water is contaminating the petrol? Inspect the sediment bowl for the mechanical or electrical fuel pump, if fitted. This may show a layer of water below the petrol. If water is seen, then you need to replace any aftermarket paper fuel filters that may be fitted, after draining off water from the bottom of the fuel tank.
2. If a paper after market fuel filter has been added check if it has become partially blocked. This is often not easily done by inspection as the filtering layer is on the inside. If the filter is more than a thousand miles old or you have been driving in dusty conditions, then replace it anyway. They are cheap enough. Fuel filters don't block all of a sudden, so a recent history of difficult starting and poor acceleration (Haha) may be useful indicators here.
3. Quite rare, but the air vent in a tightly fitting petrol tank filler cap can become blocked and restrict petrol from leaving the tank. Removing the fuel cap is the easiest quick test.
4. Simple fault, but ensure that the accelerator control linkage is correctly set-up and working OK.
If the problem only occurs in cold weather, check the choke mechanism is operating correctly. The choke butterfly valve in the carburettor should be vertical when the choke is set to OFF.
Check the idle adjustment is correct. If the engine only starts with some pressure on the accelerator pedal then turn in the accelerator adjustment screw on the carburettor to get a higher idling speed.
If the carburettor is flooding(fuel smell) the petrol level is too high the float level in the carburettor needs adjusting. Simlarly, if the float level is set too low then fuel starvation will occur.
If dirty fuel was suspected then
carburettor jets could be blocked and need cleaning out - you could try sucking the blockage through the jets by revving the engine and then partially blocking the air intake until the engine almost stalls. Workshop manuals will need to be consulted to service the jets on each particular type of carburettor.
If it is mainly a problem when the engine is hot then the Zenith carburettor can leak internally via an O-ring between the body parts, and the SU carburettor can develop a sticking jet tube which requires a little lubrication.
In an emergency situation, it's worth noting that most aerosols contain flammable solvents and so can be used as a petrol substitute when sprayed into the carburettor. Repeated use though is not advisable because the additives present could block the carburettor jets.
Fuel supply problems
There could be insufficient petrol reaching the carburettor. This could be tested by disconnecting the flexible fuel pipe at the carburettor end when the engine is OFF and manually or electrically pumping petrol into a suitable container (CARE! if engine hot).
If it is confirmed there is insufficient fuel reaching the carburettor, then it could be due to either a fuel pump problem, a blockage of the filter either in the fuel tank or fuel pump or an air leak in either the fuel line or fuel pump. Service kits are available for fuel pumps.