In short; NO, you will not be refused travel in UK due to a train being full. The only exception to this is the handful of overnight sleeper trains that require reservations.
Travelling without reservations is normal in UK, many do it all the time, trains without any seats for a whole journey are very unusual, on routes such as this you are only likely to be forced to stand when there has been serious disruption.
Train cancellations are higher than usual at present due to a number of reasons, it is best to check at the train operators website for disruptions, in this case www.scotrail.co.uk
If you want seat reservations they can be made for free by creating an account on www.gwr.com and using the Bookings - seat reservation option. when asked select anytime single or similar (NOT Advance) as ticket type for reservation, it is also fine for interrail pass. Only select 1st class if you have a 1st class interrail pass.
Cost for seat reservation is always 0, if asked for payment you are in ticket buying not reservation section.