The Lake District without a car

I looked on line and there seems to be quite an assortment of tours of the Lake District originating in Keswick, including by bike. It is a great place for hikers, if you are in good shape. Hadrian's Wall is best accessed from a town further north.

Walking is an enjoyable way to not only see the Lake District but experience the Lake District hills. However I don't know about transit to Hadrian's wall.

Local bus service is available: http://www.golakes.co.uk/travel/by-bus.aspx

Using a bus when you visit the Lake District gives you the opportunity
to see so much more. ...interactive map plots the bus routes in the
county showing their service number, the route they will take and what
you can see along the way....

http://www.lakedistrict.gov.uk/visiting/thingstodo/walking

The Lake District National Park has walks for every ability from
ambles around lakes to high ridge walks, with a bit of scrambling -
climbing over rocks without a rope.

http://www.keswick.org/what-to-do/walking-routes

Using the local buses and the Keswick Launch gives you the opportunity
to explore the whole area without needing to take your car.

Windermere also has a ferry.

In the summer months there is a minibus service between Ferry House
and Hawkshead, calling at Beatrix Potter's Hilltop House.

Unfortunately a car would be handy to enjoy a pint at the Kirkstone Pass Inn.

How much time do you have, and when will you be there?

We had a fantastic time visiting the Lake District and Hadrian's Wall last May without a dear. Did not feel we missed a thing.

First, we booked a 3-night stay with HF Holidays at their country house right on the lake at Derwent Bank, just outside of Keswick.

https://www.hfholidays.co.uk/country-houses/derwentwater/

They offer various options for guided or unguided walking, or cultural exploration. Their holidays are all-inclusive, and we thought the accommodation, food, and activities were all top-notch. Plus there is the added bonus of camaraderie with the other guests.

They picked us up at the train station in Penrith and returned us there at the end of our stay. From there we went by train to Haltwhistle, a great base foe exploring a good section of Hadrian's Wall. I can expand upon that if you are interested. But I have already lost thus post twice and want to get it finalized before it disappears again. . .

You can get around the Lakes without a car - details of buses outlined above

Hadrian's Wall will be more of a challenge

http://hadrianswallcountry.co.uk/travel/bus

Hadrian's Wall is hard to do without a car. There is a bus in the summer, but it runs infrequently. You might consider hiring a guide to drive you around. Rick recommends Peter Carney, and he was a fantastic guide when I visited. He'll even pick you up at the train station or your B&B and take you on a half-day or full-day tour.

I hope you realize that trains don't go to Keswick. The best you can do is take the train to Penrith - 17 miles away - and then take a bus.