Well, this did not cause an issue; so I started disabling things in another direction. And I found the cause (although not the solution, yet).
We have a custom post type called Products. This does not play nicely with Woocommerce Products. Once I change the name of the custom type, the site no longer times out. A bit of a flaw in our theme. Back to the developers for a fix, I think.
Thanks for your help.