Here is one to think about

I start work for a company. Sign all the usual paperwork. 6 months later, I leave, and get a P45.
2 months later, I restart with the same company, and leave after 1 month, BUT i signed nothing and get a P45.
3 months later, I restart with the same company, leave after 4 months, nothing signed, and get another P45.

Is the original contract, which I signed, still legal and in force?

Depend on the wording of the contract, be something in there that states what actions cause the contract to end.