if the light on the cord switch goes off under load, it is probably a burnt or poor connection at the ceiling switch. This is quite common as the electrical loading is very high. I would start by examining the contacts at the ceiling switch. Quite likely you will find the brass terminals are scorched or melted, the copper wire is brown and the insulation is partially melted. All the heat damaged part has to be replaced. The difficult part is that you have to cut away the heat damaged cable, and you may then find it no longer reaches, in which case an electrician will be needed because the tool required to crimp on a new length, properly, is fairly expensive and takes practice to learn to use, so is not worth buying if you will only need it once.
A new pull-cord switch will probably cost in the region of £10-£20 depending on quality and supplier. MK, Crabtree and MEM are good quality makers. It is essential to do up the connections very tight, using the exact size screwdriver with an undamaged tip, and to re-tighten them after a week or two, as the soft copper cable squashes down after time. Loose connections lead to overheating which is the usual cause of the problem you have seen.
Get a 45Amp or 50Amp switch which has the biggest contacts.
You did not mention the power ratings.
Without photos and descriptions I cannot judge if the rest of the installation is satisfactory or if additional work will be required. The current standard is for electric showers to be protected by an RCD which gives additional protection against electric shock, which can be severe in a wet bathroom.
Your fault might not be at the switch, but this is the most common cause.
