I did this journey everyday for about 3 months end of last year - well, central Croydon to St George's Hospital to be precise. I tried various methods including:
1) 264 from Katharine Street (or Park Street)
2) Walk/tram to Church Street, 264 from Church Street
3) Various buses to Purley, 127 from Purley Station
4) Train East Croydon to Balham, Northern Line to Tooting Broadway
5) Tram East Croydon to Mitcham, 280 to Tooting Broadway
The quickest was (2), but the most stress free was (1) because the bus can be quite full at Church Street. The 264 can be held up at peak times, particularly on Mitcham Common. Typically I left Croydon 5pm to 7pm, and left Tooting 8pm to 10pm.
If you're really starting at East Croydon station itself, then (4) becomes a bit more tempting. I was actually starting my journey at an office near Katharine Street so easy for me to arrive at the bus stop just in time to pick up the 264 as it starts.