Time lapse between part 1 + part 2?

I'm not sure...part 1 began February '87 and part 2 began January '88.

This article provides a pretty solid real world timeline, although it's entirely possible that the show has streamlined it a bit.

https://www.nytimes.com/books/first/b/belkin-hero.html

The election occurred on November 3, 1987.

Nick received the phone call informing him that the appeal had been denied on December 28, 1987. He's inaugurated as mayor four days later.

So they jumped forward a bit less than two months at the start of part 2.