One of my W10 64 bit desktops has a processor listed as not being supported by Windows 11. It's an Intel i5 3470 cpu. Anybody know whether I can pop in another cpu to gain compatibility or is this a new PC or motherboard exchange job?
You can not pop in another CPU. Your motherboard does not support any Windows 11 compatible CPU. The best it supports is i7 3770K, and its still only 3rd gen - Windows 11 only support 8th gen+, as of now (things might change until its release in few months though, so dont rush to buy new upgrades just yet, if you dont want to).
Weirdly enough you could get new motherboard with a Celeron J4105, which is about 85% less lowerful than your current i5 3470, and about 100% less powerful than above mentioned i7 3770K, and it would still support Windows 11, cos it's newer gen (was released in 2017).
That just goes to show that these CPU restrictions has nothing to do with whether they can actually run Windows 11 or not, whatever reason Microsoft is blocking those CPUs its not that. But it is what it is.