I have not used Joule, but as I think you will struggle to get a reply from anyone with this specific exerience I will tell you in rough terms what is involved. I would add the best people to ask are Moodle Rooms themselves.
You should then be able to hook up a core Moodle codebase (again with 3rd party plugins if your using any) and run the upgrade to bring your database up to the version of the standard code base you are using.
The upgrade process will help you out here listing "missing from disk" if it finds something in the database which is not represented in the code base, essentially telling you it is missing a plugin. At that point you would have to go back to your original code base and un-install those add ons, re-export the databse and go through the upgrade process again.
Eventually you would get to a point where the Moodle upgrade is not flagging up any "missing from disk" notices and be able to run the upgrade.
I guess you would want to know which version of Moodle your Joule site was based off, the more recent that is the easier your path to 2.7 will be.
Another option to consider would be asking Mrooms to get you back to core Moodle and continue hosting with them for a while. They do host core Moodle sites as well as Joule.
Then should you decide to move later it will be much easier to do so.
In my opinion the benchmark for service in this area was set by Remote Learner who not only released their code but have been seen to offer advice to 3rd parties on how they could remove ELIS code from an ex clients site:
That thread may give you a little more insight into what maybe involved, but of course ELIS is not Joule...
The crux of it is going to be wheter MR follow the same process RL do in working with removable plugins, and if they do how much content you have tied up in those plugins.