Hi Louise,
Assuming that the train would arrive in Euston you have two options from there.
1. Get off the train and follow the signs to the underground. Take the Victoria Line (Blue) to Oxford Circus. Change on to the Central Line (Red) to East Acton. From here it is around 10 mins walk to the hospital (and from what I remember there are signs that help you along the way)
2. Alternatively, do exactly the same as above, but get off one stop earlier at White City. Come out of the station and cross the road where there is a bus stop. Several buses from here stop directly outside the hospital so there's less walking needed.
I'd allow in the region of 30-45 mins to do the journey from Euston.
You can get a Zone 1-2 Day Travelcard for £6.60 which will cover you for the underground and buses, or just buy a combination of single tickets.
You can find helpful route information at www.tfl.gov.uk which covers the tube and bus routes.
All the best
Bhiru