Waterloo station - about 10 minutes walk.
If you mean the Royal Hospital, Chelsea - which isn't a hospital, it's almshouses for retired army personnel - the nearest main line station is Victoria and the nearest underground station is Sloane Square.
Central railway station - London - was created in 1880.
Central railway station - London - ended in 1940.
Rainham railway station - London - was created in 1854.
Sydenham railway station - London - was created in 1839.
Woodside - London - railway station ended in 1997.
Woodside - London - railway station was created in 1871.
London Fields railway station was created in 1872.
London Waterloo East railway station was created in 1869.
London Road - Brighton - railway station was created in 1877.
London King's Cross railway station was created in 1852.
Braunston London Road railway station ended in 1958.