According to the carb & jetting table on the Cambridge lambretta website your scoot should have a 5899-1 Atomiser
not a 5899-5 The 5899-5 Atomiser has a considerably larger bore diameter and will flow a lot more fuel and air than the 5899-4 which should be fitted.
The main jet for your scoot should be a 99 (0.99mm) not a 105 ! it looks like the previous owner has tried to gain some more power by increasing the fuel delivery rate which may work at higher revs and cruising speeds but will play havoc with your starting , idle and initial throttle response.
Unless your engine has been tuned I would suggest that you obtain and fit the correct atomiser and a selection of main jets to tune with . you can buy a box of 12x mixed 5mm jets for about £12 if ou shop around the net. If you can't find an atomiser at a reasonable price I've probably got one you can have
Its essential that you make sure you have no air leaks around the inlet manifold to engine joint and to the carb otherwise any tning will be a complete waste of time . We'll assume that there are no leaks .
In your case I would start by making sure that the choke cable has a little free movement at the cable adjuster on the carb, when turned to the off position. Most new choke cable inners are too short by a few mm and consequently hold the choke valve slightly off the seat all the time giving an overly rich mixture. If there is insufficient adjustment to achieve this you may have to file or saw a smidgin off the choke cable bend .
You need to set your slide height first by adjusting your throttle cable adjuster on top of the carb until you have a reasonable tickover . Don't try get it too low at first .
When the engine has warmed up properly adjust the air screw. Start with the air screw about 1/2 a turn out and adjust it a quarter turn at a time. Give the throttle a blip and then wait at least 15 seconds or so for the setting to have an effect on the tick over and initial throttle response.
Basically if your mixture at idle is too rich the engine will sound lumpy and spluttery threatening to stall maybe.You'll need to screw the idle screw IN to weaken the mixture.
If the engine speed seems to vary up and down a little by itself then its likely to be too weak at idle in which case adjust the air screw out a quarter of a turn at a time to richen the mixture .
Don't set your tickover speed too high or the engine will start to use the main jet and distort your idle speed adjustments.
If you cannot achieve a steady idle speed by using the air screw adjustment between 1/2 turn and 2x turns out then you will probably need to change your pilot jet and possibly the atomiser for the correct one
if the engine is still unresponsive then you'll need to check that your timing is set correctly and the points gap is correct if you're using points that is