Is it cheaper to leave the immersion on all the time?

In most cases, no. Leaving an immersion heater on all the time is usually more expensive than heating the water only when you need it.

Your hot water cylinder is insulated, but it isn't perfect. As it sits full of hot water, it continually loses heat through the insulation. If the immersion heater is left on permanently, it keeps switching on to replace that lost heat, using electricity throughout the day and night.

A more economical approach is to:

Use a timer to heat the cylinder before you need hot water, such as in the morning and evening.
If you're on an Economy 7 or other off-peak tariff, schedule most of your water heating during the cheaper overnight period.
Set the thermostat to around 60°C. Setting it much higher wastes energy, while setting it too low can increase the risk of bacteria such as Legionella.

The idea that it's cheaper because the cylinder doesn't have to "heat from cold" is a common myth. Heating water from 20°C to 60°C uses essentially the same amount of energy whether it's done all at once or by topping up small heat losses throughout the day. The difference is that leaving the immersion on all the time means you're continually replacing heat that's escaped from the cylinder, so you'll generally use more electricity overall.

If your electricity bills are unusually high, it's also worth checking that:

The immersion heater isn't stuck on because of a faulty thermostat.
Your cylinder insulation is in good condition.
You're not losing hot water through a faulty Temperature & Pressure Relief (T&P) valve. On an unvented cylinder, check the tundish for a constant drip or flow of hot water after the cylinder has been heating.

For most households, the cheapest option is to heat only the amount of hot water you need, when you need it, using a timer rather than leaving the immersion heater on 24 hours a day.

See this article from EasyFlow:
https://easyflow.co.uk/articles/water-heating-costs-is-it-cheaper-to-leave-hot-water-on-all-the-time