Britrail is an agency, it doesn't run trains. Virgin run the trains to Oxenholme from Euston, Transpennine Express do the leg to Windermere.
All you have to do is book a ticket on the Virgin Trains website for the whole journey. Cheap tickets do not become available before a maximum of 12 weeks before date of travel and it may be several weeks after that especially for travel at weekends. Collect the tickets at Euston for which you'll need the credit card you used for the booking & booking reference.
Depending on how far your wife can walk on the flat the easiest way is to take the direct Tube to Kings Cross and then walk 10 minutes to Euston.
Given immigration queues these days no-one can give a good estimate of how long you need but I'd allow 4 hours between landing and leaving Euston.
BUT why are you flying to London when you want to be in the Lake District 200 plus miles away? - see if you can get a flight to Manchester from home (or if you've booked add one to your existing booking) instead so you avoid all that hefting of luggage around London while worrying about missing your train. There are direct trains to Windermere from Manchester Airport.
Look for the "multi-city" option if you are returning from a different airport