How to speed up any module implementation during phased approach?

You can speed up testing process during implementation, including testing module parallel to SIT(system integration test) done by vendor prior to UAT. you define the criteria for SIT, similar to UAT, so you can get cleaner releases with less bugs in UAT, which will reduce implementation timelines when you have bug free UAT.

Also you can consider having multiple UAT environment for different modules testing to avoid dependency on one environment, which will help up testing process by parallel testing of modules in different UAT environment as compared to one UAT environment.