EXT dep 0640 LCY arr 0800
You wouldn't want to be checking in/hitting Security much after 0605, and that's after parking and getting to the terminal, or spending a lot on a taxi.
STA LCY 0800, out of the airport door at 0820 on a good day with no baggage, arrive in the City at 0845 by cab, 0900 by rail, West End by 0915. It might be worth having a look at the airport schedule for 0745 - 0815 to form an assessment of what the actual slot time might be and the likelihood of being on the stand at 0800 precisely. Marketing promises are one thing; operational reality another.
Railway is 0652 - 0915 to West End, 0652 - 0930 to City. Hmmm, the only thing going for air service is price, IF it's lower than 1st Class on the railway, and the dreadful service provided by FGW, which is getting steadily worse. ("We are very sorry for .....no catering/late/cancelled/no driver/no catering/so service/no seats/no wifi/no seat power sockets; so sorry....please don't complain or we'll have you arrested". Guaranteed to be one or more of those on any service, as you trundle towards London in repainted 1980's rolling stock.)