Another alterative would be to fly to Heathrow from a local airport and then transfer airside. I certainly would not travel down the M1 on the day - the recent closure due to a fire under the carriageway at the London end is an extreme example, but there are usually road works, accidents and endless jams caused by sheer weight of traffic. If you do have to drive down, the M40 then M4 offers a better alternative to the M1 and you can pick it up in the Midlands on your way south.