My husband and I will be traveling to England in late April for a stay of about two weeks. We plan to start in London, staying there for about a week, then move on to the Lake District and finally Northumbria, with Hadrian's Wall as one point of interest. We don't want to rent a car, so we need to choose a place to stay where it will be convenient to access some form of public transportation to various sights. We would consider guided tours but also are fine traveling on our own as long as we can get from our hotel to an attraction without too much hassle. What are some cities or towns that might be suitable? Since we also plan to fly to France for a brief stay after the UK portion of the trip, it would also be useful to know about transportation to an airport with routes to Paris.
You can fly Newcastle to Paris with EasyJet.
...... and Air France
without a set of wheels seeing Northumberland is time-consuming. If your focus is Hadrian's Wall then you've got Haltwhistle, Hexham. But then you miss the coast and the hills...
https://englandsnortheast.co.uk/northumberland-history-pages/
If you are flying out of Newcastle Airport then rather than going to Newcastle then the Metro the easiest option from Northumberland is to take the bus #777 hourly from Morpeth to straight outside the airport.
It is a 26 minute journey, first bus at 0617, so suitable for a flight after 0900 to Paris, last bus at 2247.
For anyone on a very early flight from NCL and not wanting to stay at the airport there is also a #787 bus from Newcastle overnight when the Metro is closed- at 0250, 0320, 0410 and 0440 taking just 30 minutes.
If using public transport either Alnwick or Morpeth are the major bus hub towns in Northumberland, and both are really nice and interesting places.
At Morpeth the railway station is about 10 minutes walk out of town, with a shuttle bus to the Bus Station. Handy for arrival and departure, but less handy for day trips from Newcastle to places other than Morpeth.
Likewise Alnwick, don't be tempted by Alnmouth (where the railway station is)- it is linked to Alnwick hourly by bus but many more buses run from Alnwick Bus Station to all points thus Alnwick is the much easier place to stay.
Even Newcastle would be do-able with a very frequent bus service taking just 30 minutes.
Thus there is no reason why you couldn't stay in Newcastle for both Hadrian's Wall and coastal Northumberland (frequent train to the bus interchanges at Hexham or Haltwhistle).
If doing a two centred approach Hexham or Haltwhistle are the best (but not only places) to stay for the Hadrian's Wall segment.
Given that you are starting or ending in the Lake District then Northumbria Keswick or the much neglected (on this forum) Penrith are where you want. The easiest (easier than bus to Penrith, train to Carlisle, train Carlisle to Newcastle along the Tyne Valley) way to transfer from the Lake District to the wall is bus #554 to Carlisle Bus Station then #685 from Carlisle to (Newcastle) via Haltwhistle, Hexham, Corbridge and the smaller towns in between, connecting to the #AD122 for the wall itself at Haltwhistle or Hexham.
However it is very easy to also day trip to the wall from Keswick or (especially) Penrith- it is what the locals including me do.
Finally be aware that late April will be Easter weekend then Easter week- when the schools will be on holiday, thus a bit busier. Then the first weekend in May is the May Day public holiday weekend.