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I suspect I'm looking for a simple solution that just doesn't exist but there's no harm in asking

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As your objective is no more than to be able to slot either router in place, then you really are buffering about. As you have some machines on fixed IP, these dictate your network range. change the LAN IP address on one or the other router to be on the same subnet.

That is true - your suggestion for the mask is technically better.....

But doing that will effectively combine the two networks into one - possibly with two DHCP servers allocating IP addresses to any of the PCs regardless of which network they were originally on.

If the OP is going to do that then he should use only one router - which could well be the simplest solution.

In reply to:

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But doing that will effectively combine the two networks into one - possibly with two DHCP servers allocating IP addresses to any of the PCs regardless of which network they were originally on.

If the OP is going to do that then he should use only one router - which could well be the simplest solution.

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AFAICS, the OP's intentions only extend to 1 router at a time. The network mask makes no difference to the issue of having 2 dhcp servers, which I agree could cause problems

In reply to:

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But doing that will effectively combine the two networks into one - possibly with two DHCP servers allocating IP addresses to any of the PCs regardless of which network they were originally on.

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yup. turn dhcp off on one of them.

I bit the bullet and moved the static devices onto the new router's default IP range. It wasn't so bad and I made a note of it in a short document on the server.

Make sure that you do not have static IPs in the range used by the DHCP server (unless the server controls them).

Best not to have static IP's at all. Just use DHCP, and a hosts list which ensures that the same devices always get the same IP. The "static" IP's issued by ISP's work in this manner - they are not really static at all.