Is it possible to take a Timeshift backup from a machine without UEFI and then restore it to a new machine with UEFI ??
If so, how?

To clarify - the original was installed without /boot/efi
and the new box needs that apparently

Any ideas?

Not familiar with timeshift but due to partitioning scheme differences bewtween UEFI systems and non-UEFI systems, not sure it's possible. As far as user data under /home, I don't see that as an issue but someone who has used timeshift can better speak to that.

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Probably not worth the angst - too large a likelihood that required drivers won't be there. And grub at least will need to be replaced to use the EFI aware version.
But if you insist, data are just data. And of course you can only restore what is there, so you will have to create /boot/efi manually, and let grub add its components.
Plenty of doco out there - the author has a nice how-to that covers most situations.
Some (most ?) motherboards have their own requirements as well.

If you have a running system on the target, simply restore /home - or parts there-of as suggested above.

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Thanks both, I did have the /home part covered - it's backed-up externally nightly (you know that story about deleting everything by mistake - yea, that one)

But it's just not the same heart-warming coziness as a good restore, is it?

Ah, woe is me, I'll just have to actually exert myself here.

I'm manic about backups, but I never bother with the system. Better/easier to just re-install and put back the applications I need. I always have /home on a separate partition, so that makes it easier re-installing on the same machine.
I would suggest you just do the same in need on the "new" machine. KISS.

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Same here - I always put /home on a separate SSD and back that up to my NAS, which also gets backed up...I never back up the OS though because to me it's pointless. Doesn't take me long to install and I have a backup list of the installed apps so it's easy to reinstall them from this list.