You can install a washing machine and tumble drier in the bathroom provided the plugs are not in the bathroom, there is a and regulations are adhered to. They must be at least 2.4 metres from water outlets - taps and showers and so on. They can be closer if the bathroom is too small, but must not be connected by a socket and must have a 30ma RCD.
The washing machine must be suitable for bathroom installation - check with manufacturer.
The steam from the shower shouldn't be a problem if a window is open or there is an extractor fan.
Best talk to Building Regs for guidance - certainly Part P kicks in and everything must be wired up by a Part P qualifed electrician.