There appear to be engineering works that (outbound) weekend between Bedford and Kettering, meaning that there are no direct services from London to Nottingham. You can either travel via Leicester (with no available advance fares) or via Peterborough (with some advance fares). I'd recommend the journey via Peterborough, although note that the service from Peterborough to Nottingham is on a particularly busy service - a local train on a very long route from Norwich to Liverpool!

Incidentally, don't bother looking at through tickets from Heathrow - you'll be much less likely to get the Advance purchase offers.

Is it specifically Nottingham you're going to, or Long Eaton? If the latter, ignore the above advice - you will be able to get a direct train, but it will be at the full fare, I'm afraid, due to reduced capacity during the engineering works. There may also be an option to travel from Euston to Tamworth, and then change.