The checkout page is a 'single page', so should take on your currently active theme.
I've used Community Store on a few projects lately with custom themes and I've not had to do anything to apply the theme to these pages. I can also change between themes and the cart and checkout pages update accordingly.

The only thing I can think of is that maybe your theme doesn't have a view.php file in it?

In your case, it is just using Elemental on the checkout or are you getting just the checkout fields and pretty much no styling?

Sorry, I forgot to mention that I did ensure that there was a view.php, default, left_sidebar, right_sidebar and even blank but no joy. Yes to your question about it just displaying a blank or empty template, doesn’t use my theme just displays the contents of checkout.php with no header or footer.

If you inspect the Page Templates against the theme in the dashboard do you see view.php listed as a file, as 'Wrapper for static pages' ?

I presume you are calling for 'header.php' and 'footer.php' to be included in your view file?
Any coding errors in the view file?

Any errors showing in dev console?

Any html errors when viewing source code?

If you use a 'Bootstrap' theme you shoudn't need to move/copy 'checkout.php' to the app folder.

Ahh, caching strikes again it seems. I was trying various template files (assuming view.php at the beginning) and didn't clear cache after each. All is good now!

Ahh, yes.
Same as clients not seeing updates to website when forgetting to refresh browser!

(Concrete5 caching, not browser caching)