A taxi will be expensive and could take 2 hours or more if the traffice is bad. Worst times for taxi are 6-9am and 4-7pm. The stretch of motorway you'll be using is among the most congested in the UK.

If you don't want the bus consider the Gatwick Express to Victoria, taxi to Paddington and then the Heathrow express. About a 95% chance it will take less than 90 mins from arrivals to departures. If I was on schedule I'd go for this. Cost is about £50. Quite comfortable and you get to see a bit of London too.