1.3 Heating by gas is likely to cost around 1/3rd of the amount it would cost to heat via the immersion heater. Most people regard the immersion heater as an "emergencies only" heat source.
Not quite correct, the problem is the boiler has to heat up, and the pipework boiler to cylinder also needs to heat up, so cost is dependent on how much hot water used, and what you pay for electric.
I have a iboost+ which tells me how much electric used that day, checking 3 days in a row, I got 0.54 kWh, 0.56 kWh and 0.69 kWh I only use the DHW to wash hands and the odd dish. My tariff at moment two amounts peak 31.31 p per kWh off peak 8.95 p per kWh, once I get solar sorted there will be a third amount, how much I would have been paid had I exported the power.
At the highest rate that is less than 4.83 kWh so 43p to £1.51 per week. Before I used oil, and to keep water warm to wash hands the boiler ran 1 hour 20 minutes per week, a 20 kW boiler, so around 27 kWh used with oil, to heat same tank that 4.83 kWh will keep warm with electric, so electric would need to cost 5.6 times the cost of oil or gas to break even.
I found my immersion heater had tripped, often the thermostat inside the immersion heater has a over heat trip

and it can be reset. It can trip due to thermostat being set too high, either central heating or the immersion heater, or a fault with immersion heater, so if it keeps tripping stop using it,
it can kill if it over heats also the wires under the immersion heater plate are live normally so it needs switching off before removing. It does seem at moment switched off

but can't be sure without testing. Some immersion heaters also have a switch in the kitchen, and some times feed from an off peak supply only, clearly we can't say what you have.
As to the boiler, my boiler has a wall thermostat,

and I can set how long the boiler runs to heat the DHW, there are three popular plans, C, Y, and S, there are others but not likely to find them.
This

looks a little like a motorised valve, and this

looks like the thermostat for the central heating boilers DHW. It could be a two or three port valve, and you may have a wiring centre (junction box) and some programmer

which may be part of the wall thermostat to set up when it produces DHW.
Cost electric v gas depends on how much DHW you use, regular baths or showers using the stored water then gas is cheapest, but just for washing hands electric is cheaper, and in my case with solar panels no question electric is cheapest.