No way. I would consider this if it was a flight Manchester to Gatwick, booked as one journey, so your bags are checked through and it is the airline's problem if the first flight is late, otherwise no hope.
Land Heathrow 08:10, out the front of the airport 45 mins later (with luck) = 08:55. 1 hour taxi (full fare, taxi from rank - ouch) = 09:55, if no problems on M25, big if. That leaves 1½ hours to get to the check in gate (~10 minutes), check in, passports, security and get to plane. What is minimum check in time, possibly 2 hours? No way.
Get train down the night before, from Manchester or wherever you are actually starting from, and overnight at Gatwick.