Very interesting. Amazing how much we owe, without realizing, to the early pioneers and designers of the first computers, who often prognosticated issues we deal with today.
Nah, no! Leslie Lamport is quite far away from constructive mathematics. You will for example see his explanation of logic in the TLA+ video lectures is entirely classical.
I recently have been researching Leslie Lamport's TLA+ project. Definitely in line with the ideas in this paper. https://lamport.azurewebsites.net/tla/tla.html