If you’d like to do a reservation, it may be done here: Make a seat reservation for our trains | LNER - you don’t need a pass number for this. But you are indeed trying too far in advance; it’s not possible yet. 

Also, ignore any website that charges money for reservations in Great Britain, such as ACP. They're free of charge.

Thanks so much for the replies. When will I be able to make reservations for the LNER June 2 train?

General rule in Europe: Usually 3 months in advance. Sometimes later, sometimes earlier.

There are plenty of trains between London and Edinburgh, so don't be afraid that these book out months in advance. Usually there are still plenty of seats available a couple of days before departure.

Fridays and Sundays are more popular travel days on longer distances, so check a bit longer beforehand. 

We have purchased first class global pass as we will be dragging our luggage on the trip to Edinburg. Lots of availability for those seats also, 3 months out? The process to book the reservation…..how will the LNER site know I have a global pass? Or is reservation made without charge?

They don't need to know what kind of ticket you have. The reservations can be used with any ticket and they're free of charge.

The advice from the experienced travellers in the community is to wait with activating ypur pass until the day of your first travel. Please read more about that below. 

Here is some useful information from the experienced travellers in the Community regarding both planning, reservations and activation of pass and travel days. 

 Planning

The rail planner is normally not up to date, as it only is updated once a month, so to be sure of the time table you better check the timetable and availability on the websites of the national railways. The bigger national railways, like DB (Germany) SBB (Switzerland) and ÖBB (Austria) cover several countries. 

 Reservations