Train recommendation - Southampton to Edinburgh/Waverley

If you are on a Rail pass then take an hourly Southampton to Manchester train and change at Wolverhampton or (my clear preference) Stafford.
Same train.
You need to create a free account with Great Western Railway (even though you aren't using them) to make free seat reservations.
If you are wanting to avoid London that is your only real option without multiple changes.
However be aware that Cross Country have on going strike action affecting different routes unpredictably each day, until October.
So also make back up reservations via London, just in case.

I would say Wolverhampton or Stafford is the only sane option!

But be careful. Changing at Wolverhampton or Stafford is a good idea but the onward connection to Edinburgh is only every 2 hours. Whereas journey planning software will tell you that you could change at Birmingham New Street where there is a alternative route every hour. Don't do it!!! Birmingham is a terrible station to change at and the onward trains are not as nice and are slow. The trains from Southampton to Wolverhampton/Stafford are the same type but at least it's not such a long ride.

The strike thing is very odd, clearly the union has served legal notice of a mandate to strike for that period but I'm not sure they doing it, expect possibly something at weekends. This would only affect the Southampton to Wolverhampton/Stafford part of the journey anyway. Normally you would use a departure from Southampton at around 15 after odd hours, which gives a 15 miunte connection at Stafford, but with a Railpass just reserve for the previous train (so 15 after an even hour), if it's cancelled and the next is running you haven't lost out.

And if you plan a connection at Stafford with at least one hour between trains I know an old pub with very reasonably priced food at lunchtime less than 5 minutes walk from the station....

If disaster strikes and all trains north from Southampton are cancelled there is still an option to avoid London if you start early enough. Go Southampton- Newport- Crewe-Edinburgh. Slower but some very nice scenery. The train from Crewe is the same 2 hourly service as from Stafford.

If a full strike is called, you will be given at least a two week notice. That's the law.

If train routing becomes difficult, know that Loganair offers a nonstop flight between Southampton and Edinburgh.

The railpass won't cover it but if you must get to Edinburgh that day, at least it's an option.

To me personally Wolverhampton/Stafford to Edinburgh I regard as being hourly. On the other hour I am used to thinking of it as being a Glasgow train with a same platform interchange into a Trans Pennine Manchester Airport to Edinburgh at Lancaster, Oxenholme, Penrith or Carlisle (the last can sometimes be cross platform 1 to 3 or vv). Usually a 12 minute or so connection. If it's running really tight Carlisle usually hold the Edinburgh train on the opposite platform to keep the connection.
If the connection was going to fail as the Avanti train was too late then Lancaster station has a very nice pub on the northbound platform.
At Stafford if I had unexpected time to spare in the daytime I would personally suggest to go to the very nice Alberts cafe in Victoria park opposite the station.

for the Lake District it is hourly!

As for the strike, notice for whatever may have been given weeks ago. If they are simply not working overtime it could cause unpredictable disruption. On their website it says (and National Rail doesn't)

Disruption
Due to constraints on crew availability, there will be some short notice cancellations to services:

Friday 27 June - between Cardiff and Birmingham.

No mention of anything into October, although the BBC's Bot driven travel page does say that strike action is occuring (I never have confidence in that Bot).

In trying to find a source for the doubted Cross Country industrial action which started several weeks ago I found a notice on National Rail timed at 2049 on Wednesday that it was quietly suspended, presumably for negotiations or a new pay offer to be considered. That notice also confirms that it was due to last until 25 October.

However suspended isn't the same as cancelled. As with Avanti earlier in the year the mandate is still there so action can be restarted again at very short notice. It is all very hard to predict and plan for. Avanti was on and off a number of times.

By the way the two hourly Edinburgh trains usually have a ten minute connection into a Glasgow train at the same stations. So both Scottish terminals have an hourly service from Birmingham and points beyond west, south and east. Quite a well thought out timetable.

I found the earlier Cross Country notice and it said "industrial action short of a strike" - which basically means no favours and no overtime but otherwise normal working. This actually puts more pressure on the management because most train companies rely on favours and overtime, so this causes them a lot of stress, whereas a strike causes the passengers stress but most of the managers can just shrug their shoulders.