I think I've got to the bottom of this conspiracy.
The Nottingham - Luton flow only has First Class Advances routed
East Midlands Trains Only so any itinerary with a change onto FCC at Bedford will mean these fares aren't offered. As pointed out, there are few direct Nottingham-Luton trains. However there are Standard Class fares routed
EMT&Connections so it is not surprising that you can find an abundance of Standard Class Advances for this flow.
Using a WebTIS-based site and selecting "East Midlands Trains Only" from the "other routes" part of the Mixing Deck may yield First Advances, but may involve a substantial wait at Bedford for the next EMT service to Luton.
I would instead buy a First Advance to Luton Airport and a separate ticket to go "back" to Luton. This will be much quicker and hassle-free, and only marginally more expensive.