Apple Watch and iPhone different carriers, will it effect?

Without cellular service, the Apple Watch acts like a GPS only watch. It can be paired to an iPhone that is using any carrier and it will receive messages and calls as long as the iPhone is nearby, or if its not nearby, as long as its connected to Wifi, or cellular.

If the Sprint cell service the watch was was connected to is no longer valid, then the watch no longer has any ties to Sprint either, and can be setup for cellular service with any other carrier.

Note that Verizon does have issues activating watches that have been activated on other carriers before. It is not an issue with the Watch.

As far as being paired to your iPhone, you can do that without using the cellular side of the watch. Not sure how you accomplished that, but just curiosity. All you need to do is pair the phone and watch. I don't think the fact that you had it setup with Sprint for the cellular, but I wonder if that is still valid, unless you still have the Sprint phone. The only way to absolutely be sure about your notifications for the phone is to test it. But it should work, because the watch will work with the phone it is paired to. You also have the option to turn off the cellular portion of the phone, but it is not used unless you are not near your iPhone.