Many thanks to all for your advice and suggestions. I'll try to put together a reply that addresses most of the questions:
When I joined the two wires to the (removed) gauge I did do so with a bolt and nut and made a good tight joint.
The wires to/from the gauge are heavy cables - coloured brown/yellow. No thin wires are used. When I moved the amp meters it was only a few inches and I did not need to extend the existing cables.
I have seen, on the internet, mention of a shunt. On my system there is a 'lump' of some alloy metal that is cast with what seem to be cooling fins. It measures about 5 x 4 inches and about an inch thick. The only marking on it is "Warwick Organs" and "12 Volt". The brown/yellow cables to the amp meters come from this 'lump'. Would this be a shunt?
The gauges are mounted on a wood dash.
The wires get hot ONLY with the engine running and have only recently started to get hot. This has not happened immediately after moving the gauges but some months later.
I'm thinking that a short would occur all of the time, or at least all the time the ignition is on. However, this burning is taking place only when the engine is running (charging) and, as I have indicated above, not all the time the engine has been running.
I was rather hoping there would be a simple remedy so looks like I need to bring in a specialist. However, any further thoughts would be appreciated.
Again, thanks to all