I currently use a Canon M50 with an EF-M 15-45mm IS STM Lens and an EF-M 55-200mm IS STM Lens. I am making the switch to the Canon EOS R for an upgrade and was wondering if these lenses would be compatible with the EOS R camera. If not, what adapter do I need to purchase?
Ricky is correct. You cannot mount an E-M lens on an R body directly, and there is no adapter, by Canon or a third party, for an EF-M lens to go to the RF mount.
You will need to get is preferably an RF lens, or less desirably an EF lens with a Canon EF-RF adapter. Canon now make a couple of similar lenses for the R7 and R10 crop sensor bodies: namely . If you are intending to get any of the other R-series bodies that are Full Frame units, then they would not be suitable. There is a kit option for these lenses with the RF-S 15-45 and 18-150 units. It's early days for the RF-S series, so there will doubtless by other lenses to come.
If you want a single lens to do it all, then the RF 24-240 STM lens is an excellent unit and would cover most of your range at the wide end and exceed it at the long end. On the FF bodies it will render a Field of View (what the sensor records) of 24-240, but on either of the R7 or R10 the Field of View will be 38-380mm.
The EF-M lens mount has a flange-focal distance of 18mm; whereas RF is 20mm. So to put an EF-M lens on an RF camera, it would have to be pushed back inside the camera's body. Even though the throat diameter of EF-M is relatively small, I think this is physically impossible. In general adapting one mirrorless mount to another is hard to impossible.
One day there might be diopter adapters -- adapters which use their own optics to change the flange distance -- but every review I've seen of those says they're garbage.