I used to work on eastern side of city and commute from Camberley (it was not for too long because employer relocating out west of London)
No chance of getting to LCY for 6 am by public transport from this area but car journey at that hour ok , one hour and ten minutes. The problem is getting home, pretty much two to two and half hours at any time of day. I think you are going to have to head East if youwant to do more at home than just sleep
Good luck with it
PB