No the Stagecoach ticket works only in the Lake District.
If you are here for 3 full days + 2 travelling days then you only have to make 3 journeys per day for that to pay. I would buy that ticket now if I were you then it's ready on your phone.
In London there is a flat rate bus fare. Just pay by contactless on boarding each bus. There is a daily fare cap which varies by the modes you have used, but on the bus only cap every ride each day on your 4th ride onwards is free. The computer just calculates that.
There are other threads about how to pay for the teens who my have not have their own cards. As I don't have children that is all rather beyond me in London.
There is no engineering work on Sunday but an advisory that trains around Manchester will be busy due to an event in the city.
It's unclear if you already have your train tickets but there are still advance tickets to Windermere for Sunday.
However I wouldn't buy those as you don't know what time you will actually land (as opposed to what your ticket says). If you go via Windermere the last bus to Keswick is at 6.09 pm, which means that your last possible train from the Airport is at 2.30 pm- a through train to Windermere which connects with the 5.09 bus to Keswick.
Just buy tickets at the airport station- the walk up fare is £30.80 each before railcard discounts.
Via Penrith the last bus is at 7.20pm- the train for that is a through train to Penrith from the airport at 5.10 pm, with an earlier train at 4.10 pm. If you missed the last bus due to the 5.10 being late it would be a taxi for the 18 miles from Penrith to Keswick- they are always sat outside Penrith station.
However I have just checked and the 5.10 pm train to Penrith is cancelled (no staff), so the 4.10 pm train is the last one connecting into a bus at Penrith.
Just be aware that going to London on Thursday Avanti have an overtime ban on this week. Officially all trains are running, but be prepared for on the day cancellations.
If you've not got tickets to Euston for Thursday you need to buy those today as the good fares are already gone- advance fares are now around £70 to £80 each.