Hello Darren: First two things; Paris is learning more about the real world as we speak and your thinking about the top element coming on when there is a large draw is correct for the type of heaters I’m familiar with. A problem is heaters in the UK can be VERY different than heaters in the USA. The cistern on top of the tank won’t affect heating unless there is no water coming through. You’ll likely need a meter to check voltage at the elements and also continuity of the elements. Otherwise, we’re simply guessing.
Heaters in the USA are called non-simultaneous. That is they usually have two elements and only one can be energized at a time. You may have that configuration. Even though there is a timer, you may also have the lower element controlled by the utility company to help them manage peak loads. That has been done here too.
Here, it would be the landlord’s responsibility to make sure the heater is working, so the landlord should provide you with a technician to have a look at the heater and make sure it’s functioning correctly and perhaps show you any controls and check out your electric meter to see if it’s the type that can be used for billing different rates depending on the time of day. Hope that helps 😕