I have an idea they were 113 litres nominal, but you never get that much out of them. Due to convection, the hot water rises to the top and there is always some cold at the bottom.
When heated by a boiler, they can supply around 100 litres, which is about a bathful. But your immersion heater will be 28" long at most, so it will only reach about two thirds down.
As @muchlywrong says, it is better to have an upper and a lower immersion heater. The bottom one will heat (almost) the whole cylinder, over a long period, and the upper one heats a smaller amount, more quickly.
A 3kW immersion heats water at the rate of about 1 litre per minute, so an upper one would take around 25 minutes to heat around 25 litres for washing up; the lower one would take about 1hr 40mins to heate a hundred litres for a bath.
If you buy a new cylinder, it will be better insulated, and I'd suggest the biggest one that will fit.
With your current setup, I suggest heating it for an hour or more (the thermostat will stop using electricity once it is fully hot*), pour some bath foam into the bath and as much cold as you expect to need, then run the hot until it starts to cool. Then wait half an hour and there will be an extra 30 litres of hot you can add. If you wallow for another 30 minutes, there will be another 30 litres.
The reason for the bath foam is that it forms a thick insulating layer of bubbles to reduce heat loss.
Bath oil, skin lotion, hair conditioner etc will kill the bubbles.
*if it doesn't you have a faulty thermostat, and no safety cutout, which can be very dangerous, so must be replaced immediately.