The website for booking trains within the UK is
www.nationalrail.co.uk
When you choose a journey, this site will link you to the train operator running the service. But you can actually buy all journeys from all train operators (they are obliged to sell all tickets). Don’t use re-sellers like EUrail, the trainline.com etc. They add service fees and sometimes only sell the maximum price tickets.
There are maximum price tickets available all the time, in advance or walk-up they cost the same. These tickets can be extremely expensive, and if you don’t make a seat reservation (I think at least 2 hours before the train departs from it’s first stop. You can make such a reservation at any train station but not at London underground stations, and it is free. Do not pay online re-sellers £15+ to make such reservations).
Around 10-12 weeks in advance of travel, cheap tickets called “advance” tickets are released. These are much cheaper but tie you to a specific train. Miss that and the ticket is worthless (unless it is the railway’s fault. For example, If you buy a Paris to Edinburgh trip as one ticket, and Eurostar is delayed so you miss your Kings Cross to Edinburgh train in London, that’s ok. It was Eurostar’s fault so they will find you a space on the next available service). You can sign up for an email alert to notify you when advance tickets become available for your journey for some train operators.
There are also some slightly more flexible tickets for off peak travel, which do not tie you to a specific train but do not allow you to travel at peak commuter times.
The cheapest advance ticket from London to Edinburgh is about £25. The most expensive flexible ticket (walk-up) over £200.
Breaks of journey are allowed for the more expensive ticket types (not for the train specific cheapest advance ones) but as far as I know the train journey must be completed that same day. If you want a break for several days, you would need to buy a ticket to your intermediate destination, and then another from your intermediate destination to your final destination.
I am sure the Man in Seat 61 (linked above already) will be able to explain this a lot better than I can!
You know that these journeys are not very long?
The trains from Paris to London take 2 hrs 15 minutes. But there is a 1 hour time difference. So if for example you leave Paris at 10am, you arrive in London at 11.15am.
Wander across the street to Kings Cross and it’s another 4.5 hours to Edinburgh (sit on the right on this journey for the “castle coast” part, between Newcastle-Upon-Tyne and Edinburgh. This appears in a lot of movies eg “Elizabeth”).
If you want to not plan anything you can buy various train passes. But I don’t think any of them include Eurostar (the train between England and France) because this has to be specifically booked and seats assigned (on other trains, if the seats sell out, you stand. So in your circumstances I would recommend always booking seats). Such passes cannot usually be bought once you arrive in Europe. You have to buy them before you leave home. They are not always the most economical way to go, but they allow the greatest flexibility.
In the UK a “Rail card” is different from a “rail pass”. A Rail card allows you discount of about 1/3 on all off peak journeys. So you could buy a two-together card for £30, and save if you will be making a lot of journeys.
http://www.railcard.co.uk/
If you need assistance for your mother (eg ramp up to the train - there is usually at least one step) you have to book this in advance by telephone. There are no porters at stations like in old movies. Sometimes at rural stations there are a lot of steps (and also at some London Underground stations). But mainline stations usually have step free access somewhere (this may not be obvious).
Here is the UK rail mail so you can see how all the stations join up.
http://www.nationalrail.co.uk/stations_destinations/maps.aspx
Not convinced people in the UK will understand the word “chunnel” lol! I had not heard that word before.