I've had what appear to be library issues with the Nano Every, which I gather may be related to it having a different processor to ATmega328P versions.
As I understand it, the Nano R4 runs a completely different processor again. So is library compatibility a potential issue when moving a project from a different flavour of Nano?
Yes. Any time you switch to a board with a different microcontroller, you face the possibility libraries that worked with the previous board will not be compatible with the new board. This will typically be even more likely when switching to a board of a different architecture (as in this case).
The lack of compatibility is not a guarantee. Arduino has created a standardized high level API. If a library only uses that API, it will typically be compatible with any board for which that standardized API has been implemented. Many libraries do use the high level standardized API exclusively, which allows them to work even with boards the library author has not used. The compatibility problem more often comes in the case of the libraries that have low level architecture-specific code. In this case, the library developer must implement explicit support for each target architecture. If they haven't gotten around to doing so for the architecture of your board, then the library won't be compatible.
The good new is that, although the Nano R4 is relatively new, its Renesas RA4M1 microcontroller is also used by the popular Arduino UNO R4 WiFi and UNO R4 Minima boards, which have been around for some years now. This means the library developers have had a fair amount of time and motivation to make whatever changes might be needed to support these boards. And a library that is compatible with the UNO R4 boards will be equally compatible with the Nano R4.