The neon indicator just shows that the power is turned on - it does not show that the water in the immersion heater tank is up to temperature when it goes off - the indicator will be lighted all the time the switch is in the on position.
You do not say if your water is getting hot or not - if you get hot water then things are OK.
Your comment about the "outgoing black and red" is confusing and unusual - it suggests that the immersion heater might be on a ring circuit - my understanding and experience is that they usually sit on a dedicated feed.
I found this answer in a discussion...
Immersion heaters should be on their own dedicated circuit, 16a breaker and a 20a DP switch, not a fused spur. Having said all that there are plenty around wired the way your's is, it was the usual way of doing it some years ago.
Good luck...
March 2015