Heathrow is roughly 145 road miles from Nottingham. By car it is typically 2 hours 30 to 3 hours 15 depending on the M25. By rail it is St Pancras plus an Underground or Elizabeth line connection. Here is an honest comparison.
Door-to-door time
A direct taxi runs door to terminal in a single leg: usually 2 hr 30 to 3 hr 15, and closer to 2 hr 30 for overnight departures when the M1 and M25 are clear.
By train, Nottingham to London St Pancras is about 1 hr 45 to 2 hr, then roughly 15 minutes on foot and Underground to Paddington or Farringdon, then 30 minutes on the Elizabeth line. With connection buffers the realistic total is around 3 hr 15 to 3 hr 45 — plus getting to Nottingham station in the first place.
The early-morning problem
Most rail services cannot get you to Heathrow for a 06:00 check-in. The first useful departures land you at the terminal mid-morning, which means an overnight hotel for early flights. A taxi collects you at home at 02:30 and removes the hotel from the equation.
Luggage and groups
Rail costs scale per person, so a family of four multiplies quickly, and Underground connections with four large cases are hard work. A fixed-fare 6-seat or 8-seat MPV carries the whole group and the luggage for one price, with the driver handling the cases.
When the train wins
For a solo traveller with hand luggage and a flexible midday flight, the train is often the cheaper option and we will say so. For families, early departures, late arrivals or anyone with real luggage, the door-to-door transfer is usually the better call.