10 Days Over Christmas

The Stansted Express runs to Liverpool St station in London. For the sake of convenience somewhere around there would be good. Liverpool St is on the Circle line and a few others so it should be easy to get around. Don't rule out easyjet flights to Gatwick though - they can sometimes work out cheaper.

As for daytripping/travelling, depends where you want to go. If there's a couple of places you want to go clustered in a similar area e.g. Cardiff and Bath, then by all means stay overnight.

Your lodging budget is tight for London unless rates drop in late December. I traveled there in early September and decided I wasn't going to do out-of-town day-trips after paying London hotel rates. I'm not sure Bath is much cheaper, though, and in general I found English hotels more expensive than comparable properties in France.

I was very happy with my stay at Vincent House. It mostly rents rooms for long-term stays--I guess primarily to students or visiting faculty, but it sometimes has space available for short-term visitors. The rate will probably be at or just above the top of your target range; a filling breakfast is included. The location is good: one long block from the Notting Hill Gate tube station (Circle Line and Central Line, the latter allowing good connections to the other tube lines) and walking distance to the Portobello Road market. If you hit good weather and don't mind a 35-minute walk, the V&A is within range, as are the nearby Natural History and Science Museums.

These are places I visited before ending my trip in London, with base cities in bold:

• Bristol/Cardiff/Bath (tours to Cotswolds run from Bath, but I'm not sure about that time of year)
• Oxford/Bletchley Park (tours to Cotswolds also run from Moreton-in-Marsh, a very short train ride from Oxford; again, don't know what is available in December). Oxford has two good museums--indoor sights that would work well for a winter visit.
• Brighton/Lewes/Arundel

I've never been to Europe in the winter and would probably just hunker down in London, scurrying from museum to museum, if I opted for England in December. My recent London visit was 10 days, and I covered less than 50% of the museum exhibits I wanted to see. You're unlikely to run out of things to do.

Edited to add: I'm 65 and also a solo female traveler, if that helps.

Edit #2: I yield to no one in my hatred of trekking long distances with a suitcase, but on a long visit like this, I think it's more important that your lodgings be convenient to the places you want to see than be very near airport transportation. In particular, if you think you'll be going to the theatre often in the evening, consider ease of transport from the theatre district and/or the National Theatre to your lodgings.

My first thought and suggestion is to rent via AirBnB or VRBO.

Look for an apartment or a room in some one's house, flat etc. Some small restaurants and pubs will be open but Christmas is a day when London shuts down. NO public transport so if you are outside of the city center you need to consider whats near by.

If you were to rent an apartment or room you could cook for yourself. A good money saver.

If you were to rent a room in someone's flat like I did last year I had access to their kitchen. The owner didn't arrive home from China till late christmas day so I had the place to myself. Private bathroom. Bed was comfortable. Host lovely. About 15 minute walk from the closest tube station.

A bit out of the city center but that was my choice as it was a neighborhood I'd never visited. I'll PM you it's locale.

Otherwise check availability at the Vancouver Studio Apartments and check it NOW!!! Have stayed their more than once over the years. Never been disappointed. Bayswater neighborhood is vibrant, full of cafes and shops and 2 different tube stations to chose from.
Can walk to Hyde Park, Portobello market.

Anyway on Xmas It's quiet and calm and rather lovely if you are a walker.

December is not the best time for the UK, cold, grey, damp and short days. But it is Christmassy. It will be a lot colder than Seville.
Stansted is half way between London and Cambridge, and has hourly trains to Cambridge, so you could do a split trip with 2 nights in Cambridge.
Other places to consider (you could do a loop by train): Lincoln, York
A day trip by train from London is possible to Oxford, Canterbury, Winchester and others. Remember a day trip in December requires leaving in the dark and returning in the dark.

I am not sure how much you mean by "$100 - $125 range", as they don't use '$' in either London or Seville. If you want cheap and basic London hotels, try one of the Premier Inns chain. Modern hotels with many in central London: http://www.premierinn.com/gb/en/home.html
Or try this one: https://www.imperialhotels.co.uk/en/president

If you go to Salisbury, there's a shuttle bus that runs to Old Sarum and Stonehenge. You could spend 2-3 nights with a full day to see the sights of Salisbury itself and the other day to use the shuttle. However, I would say to avoid Stonehenge and the entire Salisbury/Wiltshire area at the solstice unless you're a diehard wanting to experience the solstice -- which I'm guessing you're not since your original post didn't mention this.

I was in London at Christmas last year. Most of the time it was between 36 degrees F (cold! hat/gloves) and 45 degrees F. We had a bit of cloudiness, about 1 hour of drizzle, but mostly sun. Overall terrific weather for touring. Sun does go down by 4:30 but that didn't stop us...and makes for a lovely background for the holiday decorations on Oxford Street.

Please note that it would be a good idea to look at the websites for the places you are thinking of visiting, paying attention to "holiday hours". Some places in Oxford/Cambridge may be closed for break, or may have limited access to those university buildings that are open--not just Christmas Eve/Day/Boxing Day but adjacent days as well (which happens in the states too--students go home and colleges have limited open buildings).

In London, if you enoy history/architecture and walking, and haven't checked it out already, look at "London Walks". Very professionally done, a diverse set of walking tours, well organized and affordable.

Enjoy your trip.

A few things to bear in mind:

Many tourist attractions, museums etc in London and elsewhere will be closed over the Christmas holidays - often 24-26th December - so please doublecheck opening hours before making plans.

Rail companies tend to use the whole Christmas/New Year period for major engineering work as so many commuters are off work and thus it's the quietest time of the year. This means that if you're taking day trips by train, you will need to check train times very carefully - don't assume they'll be running a normal service. Schedules may be published quite late. (You won't find any trains on 25th or 26th December.)

There is no - none, zilch - public transport in London (or anywhere) on Christmas Day itself. Many restaurants will be closed. It's the one day of the year when the country almost grinds to a halt as people spend time with their families. You'll need to make your plans carefully - find a restaurant that's open and within walking distance, go for a nice long walk in an oddly quiet location. I'd advise staying as close to the centre as possible.

It gets dark that time of year at 3.30pm. Daylight hours are very short. This may effect your plans if you're interested in rural sightseeing, for example.

$100 - $125 is a low hotel budget. £100-£125 would be better - that would get you a comfortable budget hotel. Bloomsbury is a nice area with some good budget hotels - look at somewhere like St Margaret's Hotel or Arosfa Hotel.

London is lovely in the run-up to Christmas with great lights in the shopping streets and a buzz about the place. Cambridge is a great suggestion either to stay for a couple of days or for a day.

Little-known tip is Pax Lodge, the hostel run by the World Association of Girl Guides and Girl Scouts, for female travelers who want a friendly, safe place to stay. Prices are good, dining hall in the building if you want, residents have a kitchen they can use, or George Pub around the corner. It's next door to Air Studios in Hampstead (famous recording studio that has done Harry Potter films, long list of famous movies' sound tracks) in a convered big church. I lived and worked there for months with my grown daughter back in 2011. They rennovated recently so there are new beds, etc. They have an elevator, gift shop, and front desk sells discount tickets for attractions. Corner bus stop is Rosslyn Hill, and it's on routes 24 (goes right to big sights) and 268 (to Waterloo station and more) plus a couple of others. Belsize Park tube stop down the hill, across the street from a little grocery store.
If you book directly with Pax Lodge, it includes breakfast (when on booking.com I think it does not include breakfast).
Easy bus rides to nearby Camden and Euston and King's Cross (and St Pancras, British Library's treasure room).
Often it's less crowded over the holidays, and there's staff that stays there who are friendly and doing extra celebrating and decorating.
They used to have a slight discount for Girl Scout/Guide members (and Boy Scouts too), but I'm not sure if they're still doing that.
If you're not a registered Girl Scout or Guide member of your country, it costs like maybe $5 more (so it can be worth it to call your local council to get registered as a member for $20 or so, not a big deal either way).
https://www.wagggs.org/en/our-world/world-centres/pax-lodge/book-now/

Usually the other guests are older scouts and leaders, and often some older people who are there for the hospital services down the hill. Activities and meetings get over with fairly early in the evening, and the groups that I saw were fun and well-behaved.
Hope it helps :)
-Alison