First of all if you just installed Solus anyway, I would strongly recommend just doing a fresh install from the Budgie ISO instead of replacing the desktop environment (which will leave you with lots of additional packages you don't need, possible configuration issues, etc.)
Now, if you want to continue down this path (again, not recommended), there should be an option on your login screen to switch to Budgie, provided you installed it correctly (i.e. following these instructions)
If I'm not mistaken (can't check at the moment) you have to click on the little MATE logo to get all the available options for different sessions.