This is certainly possible. There are a few options. There are sockets which include a timer. Anything plugged into this socket will turn on and off according to the schedule set. The timer takes up the space of one socket leaving the second socket available to plug in your towel rail. If you only have a single socket then a spur type timer would be a better option. A qualified electrician can replace your single socket with a timed spur and remove the plug and wire the towel rail directly to the timer. There are also plug in timers which you can purchase and this will achieve the same result but will be less attractive.

Yes of course

There are aftermarket timers that can be fitted but it depends on the age and manufacturer. It may be necessary to replace the towel rail with one that has an option for a timer.

You say you have a switch outside, but is this a switched fuse connection point? If so, it could be replaced with a switched fused timer, such as a Timeguard FST24

Yes. You would require a fused timer spur similar to what you would use for an immersion heater.

There are various options that can be used, I would recommend a qualified electrician to have a look at your installation & come up with a solution to your question.