Not really, I would have expected the red ones to be Live because you can still get the heating etc on with the Horstmann 24hrs override switch. The black Neutral wire would not be expected to light up the tester.
You need a Neutral to complete the circuit which drives the motorised clock dial around and it looks like that is what's missing.
A multimeter would let you measure the voltage between red and black, something you can't really do with a neon tester.
It's possible one or more of the old backplate contacts are corroded and not making a good connection to the main unit so, if you feel confident, carefully swapping all the connections over to your new backplate might fix it.
Unfortunately it may turn out to be a problem in the cable or at the far end. Without a way to measure the voltage you won't know where the Neutral is getting lost.