You totally do not need apps to travel by rail in the UK.
It is often made to sound far more complex than it is. Yes there may be multiple train companies but each can sell each other's tickets so buying from multiple websites just isn't needed.
What you need is a Two Together Railcard. With that you get 33% off all rail fares, (on Monday to Friday travel after 9,30am, anytime at weekends). This way you pay £35 for one railcard, not £35 for each of two Senior Railcards.
You can buy your tickets saying you have a railcard, but can wait until you are in the UK to buy the physical railcard at any staffed station. It doesn't have to be held on an app/a device.
The easiest website (yes website, not app) to buy from for overseas visitors is the LNER site.
The easiest method is to opt to collect physical tickets from ticket machines- when you buy on line you will be given a collection code which you put into the machine along with the credit card you used for purchase (merely to prove your identity).
To Hampton Court and Windsor just pay on the day at the station. Real tickets (and many people still use them) are the same price as paying by contactless methods on those journeys.
If you are willing to be tied to specific timed trains the websites will always offer you the cheapest most restrictive fare first.
For York, Canterbury and Southampton tickets are available usually from twelve weeks beforehand, but are also available until far closer to travel as well. Unless you are trying to sweat on saving every possible nickel you don't have to be buying so far before travel- the sweet spot I find is around two to four weeks beforehand.
No UK daytime train can sell out- they will always have room (if sometimes standing room).
Yes you can add complexity and mystique to the process if you want to (and many do) but it is not necessary (at least on this itinerary) to do so.