Luton is not in London, it is in Bedfordshire. Whey they call it London Luton is beyond me. Let's say your flight arrives on time at 12:25. You then have to disembark, wait for luggage to be put on the carousel, collect it, clear customs and exit the airport. In my experience, this takes a minimum of 1 hour.
So you get on the Heathrow express (every 15 minutes) so lets say you catch one straight away at 1:25. You arrive at London Paddington Station at 1:40pm.
You then need to buy a tube ticket to London Kings Cross St Pancras. You can go via the circle line or the Hammersmith and City Line.
If you have luggage, allow extra time as there are stairs at the station. This should take around 30 minutes but I'm adding on 5 minutes for you getting off the Heathrow Express, running to the tube, buying a ticket and running down the stairs.
This gets you to Kings Cross St Pancras station at 2:15. St Pancras is a walk from the main underground station, follow the signs (again, there are stairs but escalators in the station). Arrive at St Pancras at 2:20.
There is a train at 2:30 which goes to Luton Airport Parkway, arriving at 2:50.
You then need to change trains for the East Midlands service to Nottingham at 3pm, getting to Luton Airport station at 3:06.
You might want to book your tickets in advance as there are often queues at St Pancras for tickets, you can purchase a through ticket to your destination.
Once you arrive at the station at 3:06, you then have to walk to the airport and terminal building for departures. You should arrive at your check-in desk at around 3:15.
Depending on where you are flying to from Luton there is a minimum check in of 1 hour (boarding starts 45 minutes before flight leaves) so assuming you have factored this in and you can run very fast and no trains are delayed you just might catch it with luck on your side.
http://www.lutonairportinformation.co.uk/luton-airport-INF-passengers-guide.php
(check-in information)
However, this is a very very tight schedule and I would not attempt it myself and I know London very well. Don't be tempted to get a cab as London traffic is awful and an hour later you will still not even be out of London. I would seriously look at travelling home by train if you can. There are express services to most major cities and from London Euston to Manchester it is only 2 1/2 hours.