I don't have to interact with both Jane and Mike, but .
If you are interested in your friend's ex, go slow and steady.
We know that you've just got your golden opportunity and you can't just waste it.
Dodging the issue and sweeping it under the rug won't do anything to reduce tension, it will only allow it to fester. Take this to your conversation, too, and discuss ground rules and expectations for your relationship with Mike going forward.
Maybe Mike feels just as awkward about it as you do and is planning to remove himself from the friend group so you don't have to see him - you won't know until you talk to him. If the talk goes well, fantastic, you figured out how to act around each other in the future.
I would like to keep in touch with these friends and keep hanging out with them regularly.
I want to keep my friends (including Mike) and basically not address the fact of them dating. It's good to recognize they can do what they want, but your feelings are also valid. This will help keep you on track if things get difficult.
That would be awkward for everybody, so I am looking for a solution which would do the least damage to friendships and still keep my face. If these people are your friends, they should care about you too. Try to stay focused and calm during the conversation - remember that he didn't do this to hurt you - and don't be afraid to take a step back if you need to.
My issue is that given our history, I have no idea how to interact with them when it comes to it. So if you want to clear the air: contact Mike and ask if you can talk. If your goal is just "be not awkward around Mike", what would that look like for you? Perhaps this means "we won't talk about Jane", "we will acknowledge each other at events and act civilly, but I won't expect him to talk to me one-on-one".
I found out through a mutual friend that a few months later, she started dating another of our friends, Mike.