Our current Access database operating system has the standalone full Access 2010 on my PC, which has both front end and backend installed
The other users have Access 2010 Runtime installed on their PC’s which then allows them to use the front end only
If I look at upgrading to the latest Access version, is there a Runtime equivalent and is it a smooth upgrade from 2010?
Most of the time upgrades are relatively smooth, yes. However, one should ALWAYS start out with a single user (probably you) and thoroughly test your Access relational database applications before releasing it to other day-to-day users.
One thing to be very cautious about is the 32-bit/64-bit difference. Your Access 2010 is, no doubt, a 32-bit application. Microsoft now preferentially installs the 64-bit version of Office if you don't specify otherwise. That can introduce problems. For that reason, most long-term Access developers would recommend sticking with the 32-bit version of Office 365 as well. However, it's probably a good idea to at least give some thought to moving to the 64-bit world.
The two primary issues with 64-bit Office will be any ActiveX controls used in your 2010 accdbs or any APIs used in them. There may or may not be 64-bit versions available for any such ActiveX controls you have used. You'd have to replace them one way or the other, i.e. with a 64-bit version or something else. APIs can be modified to run both under 32-bit and 64-bit Access. However, even though the modifications are not difficult, they do require thorough understanding of what's involved.