There appear to be three separate things becoming muddled together here: the immersion heater in the cylinder, any electric booster built into the Bosch indoor unit and the solar controller connected to the upper immersion. They may all produce heat electrically, but they serve different purposes and may be controlled completely differently.

Some Bosch heat pump systems have an integrated electric booster in the indoor unit. That can assist the heat pump when required and, depending on the configuration, may contribute to space heating as well as hot water production. That is not the same thing as an immersion heater screwed into the Telford cylinder.

The cylinder immersion is normally there for backup, direct hot water heating or the periodic high-temperature cycle. It cannot directly add heat to the radiators. As James says, its useful role during very cold weather could be to heat the domestic hot water instead of the heat pump, leaving the heat pump with more time and capacity to heat the house.

However, from what you have described, it is far from clear that this mystery switch currently does that.

If the upper immersion is wired directly to the solar controller and the lower immersion has no wiring connected, then the switch cannot simply be assumed to operate either element. It might supply a contactor or part of the control system inside the wiring centre, or it might be a redundant supply that was never completed. The only reliable way to know is for someone competent (good luck with that) to trace the cable, inspect the wiring diagram and test what becomes live when the switch is operated.

I would not simply turn it on and leave it on based on the installer saying it is "a booster". After 18 months, you should have been given a clear schematic and commissioning record showing exactly what every supply, heater and control does. The fact that Telford, the installer and two service engineers have given you different answers is not reassuring.

More importantly, the heat pump struggling in cold weather does not automatically prove that it lacks capacity. It could be capacity, but it could also be the heating curve, restricted flow, thermostats or TRVs shutting circuits, poor balancing, hot water scheduling, a disabled booster setting or the unit not being configured to run continuously.

I would ask the installer to confirm in writing:

1. What the switch actually supplies.
2. Whether the Bosch has its own internal electric booster and whether it is enabled.
3. Which immersion element is intended for backup and which is controlled by the solar diverter.
4. Whether the heat pump is designed to meet the house heat loss at your local design temperature. 
5. What temperature and flow rate the system was commissioned to deliver.

In the meantime, please post the exact Bosch heat pump and indoor unit model numbers, the Telford cylinder model number and clear photos of the switch, wiring centre and both immersion connections. That should allow us to get much closer to a definite answer, and I can ask some credible installers to take a look. 

Using the immersion for hot water during the coldest periods may well be a sensible workaround, but first we need to establish whether you genuinely have a capacity problem or whether another installation and commissioning issue is once again being disguised as "the heat pump struggling".