I've never had the definitive answer to this to be honest. I remember that the (now defunct) Midlands Electricity Board (MEB) always used to promote immersion heaters to be left on . But then we all bought our power from them in this area so they would say that wouldn't they?
As said above , there are some important criteria, a modern cylinder factory lagged , Economy 7 tariff and your particular usage. If you are filling say , two baths per day ,that would be expensive and an electric shower would be far cheaper to run. A bath will empty your cylinder .
If the immersion is just for washing up and a morning wash I'm thinking a time switch may be save money, or a selectable top element but heating up from cold is the most expensive , I'm taking a guess at 2.5 hours @3KW before the stat switches off.