How much would it cost to install a new electrical outlet in the washroom?

£73.08, including VAT and regular biscuits and coffee.

1 kajillion dollars!

No matter the cost, the most important thing is to make it right. Install a GFI receptacle. Be sure that the amperage of the electric circuit complies with the required amperage of the new device, for example, having a 7 amps device to be installed, the electric circuit must be of 15 amps. You can check the amperage 0f the electric circuit just by looking at the fuse or circuit breaker than controls it.

If the device requires 15 amps, you must connected it to a 20 amps electric circuit.

Use the proper connectors from the box you will use to pull the electric, use the proper size of wiring, use the proper boxes.

About prices, your question is in the limbo, because you must give information about the country, and specific area, due to the charges change according to the geographical location.

Do you have any electrical supply near the toilet like a light fixture? The easiest thing to do would be to run a line from the light fixture to create an outlet near the toilet. The light fixture is almost certainly on the GFI circuit already.

I don't think is code approved to pull electric from a light circuit to feed a receptacle. I know that many people does it, but in case of causing a fire (bad installation) the insurance company won't pay a single dime.