You're assuming that you will actually land at 0630. Is that a safe assumption? You're unlikely to land early (noise restrictions before 0600) and the 0600 to 0800 window is peak arrivals...
You need to get through immigration and out of the airport. As above, you're in the peak arrivals window.
The National Express coach is scheduled to take an hour (you're in peak rush hour on the M25) and most days, that'll be an accurate journey time. But sometimes (perhaps one day a month) there'll be traffic meltdown and the journey will take much much longer.
BA will close check in at LGW at T-45. Even if everything has gone swimmingly, you'll be very close to that cut off.
Way too risky for me.