Water – Hot water stops running intermittently

I've not used a system like you have, but I'll give you some information based on what's common for heating systems in the UK.

As you've discovered, the immersion switch immediately switches the heater on. It's usually used as an override when you want water to start heating outside the timed period. (I'd hazard a guess that having the immersion on for 10 minutes won't be long enough to heat sufficient water for a shower).

You'd usually set the heating on timed, but it will take a while to heat up the full tank of water. For example, if you get up at 07:00 and have a shower, you might need to set the timer to come on at 06:00 to give you a tank of hot water ready for when you get up.

It's common to set the timer to come on twice a day, so you get hot water in the morning and again in the evening - though you'd adjust that dependent on your usage pattern. It might take a little experimentation to find out when you need to set the timer to come on to give you the hot water you need.

If the T&P valve is flowing quite an amount of water, it sounds like something is amiss with the system.

Regulations, Regulations, Regulations. However, your existing boiler should have been installed under some previous regulations so changing over to a combi and getting rid of the tank stuff shouldn't be a problem for a qualified gas safe engineer. One of the safety aspects is "Ventilation", but you already have a boiler in place so spend a bit of money on getting legal/professional advice and then invite 3 companies to quote you. Good luck