... offside and nearside "front lower suspension arm ball joint dust cover deteriorated, but preventing the ingress of dirt"!
That's the ball joint attaching the lower suspension arm to the wheel upright. The good news is that new suspension arms are relatively cheap and someone who knows what he's doing should be able to change them quite quickly.
The handbrake probably needs some attention.
Look at the rear brake calipers and ensure you have the two return springs ( 1 each caliper ) that pulls the handbrake actuator back to it's rest stop. If you don't have the springs,you can search on here for lambretta. You can see the spring in this photo -
( just above the 'PS' in the QPS logo ).
With the springs in place, slacken off the handbrake completely ( there an adjuster nut under the handbrake lever ) and then go through the handbrake adjustment routine :
Unclip the handbrake cables at the rear calipers and pump the brake pedal ( 10 times it says in the manual ) to ensure the pad is fully in contact with the disc. This will reset the handbrake mechanism.
Check the handrake actuator lever. It should be be fully against it's stop. Reattach the handbrake cables, pull the handbrake lever up 3 clicks and start tightening the adjuster ( at this point the wheel needs to be on the car and off the ground and it's better if both wheels are off the ground so you can check there's no imbalance between the two sides ).
Try turning the wheel by hand. If it turns freely, keep tightening the adjuster and testing the wheels. You should keep going until you can no longer rotate the wheels by hand. Release the lever and check the wheels turn freely.
Go and park on a 16% or steeper hill. If it holds the car and the lever doesn't pull more than about 5 or 6 clicks, you should be OK for the MOT.