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.
Why is knob and tube top priority to replace? It's quite safe if left alone. You only need to replace it in visible areas: the stuff behind the wall doesn't need replacing.
You might want to add some circuits in certain rooms, but you don't have to remove the old ones.
No upgrading the radiators (from your description they don't sound like baseboard heaters) will not increase efficiency. Forget about "huge efficiency gain" - it's doubtful you will have a gain at all.
What might increase efficiency (somewhat) is balancing them. An old house like that probably has only one zone, so you need to adjust the settings on each radiator so that the house is balanced - that some rooms do not end up hotter than others.
This can take a long time to do. To start with, do nothing - someone might have already done it for you. Then see if the rooms without the thermostat are hotter or colder than the one with. If all the rooms are hotter/colder, then adjust the radiator near the thermostat the opposite way.
Now that you have it mostly set, start adjusting each room based on the temperature. Wait like a day between each change and be aware the correct setting may vary based on the outside temperature, which is annoying, so you'll just have to find some reasonable midpoint.
If you want to upgrade anything it should be the insulation on the house.
Best Answer
That baseboard radiation will work great on a 2 pipe steam system or a forced hot water heating system but is not recommended for use on a 1 pipe steam system.