Glad to hear that you are not doing it yourself!
As i said
You have a voltage ratio of about 20:1. (240volts > 12volts).
So (in general terms) the primary current will be a 20th of the secondary current.
Your secondary current (you calculated correctly) as 20.833amps.
A 20th of that, as Bernard has stated is just over an amp, so your existing lighting circuit may be able to cope with that, depending on what it is already driving.
But your registered electrician can help you plan all that. I suggest you leave the supply of lights to him too.
There are a number of considerations: SELV may be required, they may have to be IP44, maybe better than that, depending on bathroom layout.
There's building regulations to comply with:
Among others, parts B, E, L and P will apply to downlights.
Then you/he will have to consider how you stop the heat from the room escaping into the loft. Do you have lots of insulation? Now you are going to make a swiss cheese out of your ceiling. You can't just put insulation over your downlights.
And all that steam, that will get drawn up into the loft space, and the moisture will rot your roof timbers - unless the lights are the right ones.
Oh yes, and lets not mention the fact that you will be burning 250watts to light a room that can be adequately lit with a 16watt low energy 2D light.
Still want to go ahead?