Had the same car. The Lucas electric fuel pump would decide to stop working at will. I kept a large screwdriver to give it a whap with the handle to get it going again.
I suspect that is what is wrong with my MG Midget.
On the flip side, from spring 1966 to summer of 1970, my daily driver was a 1965 AH Sprite, including Iowa winters. Never had a Lucas problem. Was one of the most dependable cars I ever owned.