Advice on 4 day trip from Edinburgh to near Heathrow

The most obvious place to stay and take a few day hikes in beautiful countryside would be the Lake District. You can take a train from Edinburgh to either Penrith for a bus connection to Keswick or to Oxenholme for a train connection to Windermere.
Keswick is a small town rather than a village with a range of accommodation. Windermere is likewise and a little more developed but there are a range of fine hotels within a short bus or taxi ride from there that will put you close to smaller places like Ambleside and Grasmere.
When you’re done just get back to Penrith or Oxenholme for a direct fast train to London Euston and onward transport to Heathrow.

Specifically regarding a gym and spa, the Keswick Leisure Pool (down by the old railway station) has a gym.

I don't think there is a spa actually in Keswick but the Underscar Hotel and the Armathwaite Hall Hotel have spas, as I think does the Castle Inm. They are all beside Bassenthwaite Lake on the X4 bus route from Penrith to Keswick to Workington. That runs every two hours, alternately with the X5 down the other side of the lake.

You could stay at any of those hotels, all of which have easy access to the High Fells- especially the Underscar, but still have easy access to the facilities of the nearby towns.

If you want more of a villagey theme try the Embleton Spa Hotel, which has all you could want in fitness and spa. That is a bit further on the X4/X5 bus routes towards Cockermouth. A great place for a walking holiday, including walks out towards Lorton and Buttermere, which are maybe less well known.

The Lodore Falls Hotel also has an extensive spa. I didn’t mention it as it’s not in a village but it is on a bus route and the Derwentwater ferry(?).

Down Windermere way the Langdale Hotel would be a good place to stay with their spa and fitness centre. That is by Elterwater and Chapel Stile, with as much good walking of all abilities as you could possibly want. I am sure you could stay in cheaper places in the village and use the hotel facilities, even if you had to pay.
They also have self catering lodges there.

At Grasmere, in between Windermere and Keswick on the 555 and 599 (open top) routes you have the Daffodil Hotel with full spa and fitness facilities on the village outskirts- the bus stops right outside. Grasmere is certainly a village and totally different by night when all the day trippers have gone- but lots of places to eat in the village, and of course the Gingerbread (!) and the Wordsworth connections.

The Lodore Falls Hotel does indeed have it's own boat landing (just across the fields) on the Derwentwater Launch route, about half way round the route, whether you go clockwise or anticlockwise).
As it's name suggests there is a pretty good waterfall behind the hotel. Many people like me still informally call, it the Swiss Lodore hotel- it's original name.

My wife and I had a short post Christmas break there this year and in booking it I spent some looking for the Swiss Lodore!

There was a public house on the site since 1792, which was extended in 1870 to become a hotel. But it didn't gain the name the Swiss Lodore until 1947 because the then new owner had a Swiss wife. It apparently (according to my father) was quite Swiss Chalet style inside. Eventually it was a Hilton hotel for a few years around the year 2000, but was purchased by Graves Brothers in 2004. They gave it it's current name and built the Spa.
It is now part of a chain-Lake District Hotels- owned by Graves- all priced at the top end of the local market.
When it was built the spa was a bit of a carbuncle but it has mellowed well into the landscape.
Graves are local entrepreneurs in the hospitality and entertainment business, and at one time owned almost all the cinemas (movie theaters) in the area- now down to just two. The others have either closed or bought themselves out of Graves' hands (like the Alhambra at Keswick).
Until relatively recently the general public had to pay to go to see the waterfall.

The hotel is now well beyond my budget- especially as it is only 30 minutes drive from home.

Thank you everyone for the great suggestions. Plenty to work with!

I've been racking my brains to find a spa hotel for you near Hadrians Wall, so that you can do the northern most parts of the Pennine Way.
The wall may not count as challenging walking but the Pennine Way from Alston to Kirk Yetholm (especially the last day) certainly is.
That last day is much closer to 10 hours and is tough going in seriously remote terrain.

I have tried a number of thoughts with places that might have spas, and have drawn a blank, at least if you don't have a car.

The obvious choice, with a car, would be the Gilsland Spa Hotel but, that is useless on the bus.

And I can't think of anywhere else in Northumberland (accessible by transit) that has what I would call challenging walking. I was thinking about getting off at Berwick, Alnmouth or Morpeth on the east coast and using buses to do something from one of those. Even the coastal walking (if you based at, say, Bamburgh or Seahouses) doesn't qualify for that word.

Further south, in East Yorkshire, either in the Wolds, or the Yorkshire coast, say Whitby to Scarborough could qualify.

The other thing to mull over is the far eastern section, over the North York Moors, of Wainwright's coast to coast walk.

Picking up on isn312c's comment - I'd thought about Slaley Hall Hotel and Spa about 8 miles south of Hexham, but again its going to be difficult without a car (train to Hexham and then taxi). Although there is a lot of walking around, (have a look at the OS map here) and it does give access to what is marked on the OS map as 'A Pennine Journey'. This follows the route of a journey made by Alfred Wainwright in 1938. I'm not sure whether this would qualify as a challenging 4 hour walk though...

Thank you!
We are now going to rent a car to make it easier.
If you have any suggestions with this change we would love to hear.
We will look into the ones currently mentioned.

Having a car doesn't change the Lake District recommendations but makes them a lot easier.

From the Underscar, especially the walking is straight out the front door. From the Lodore you will probably need either a boat across the lake or an open top bus to the head of the valley- but are spoilt for choice. From Armathwaite Hall or the Castle Inn just a short drive down Bass Lake to Dodd Wood car park for Skiddaw or the head of the K2T rail trail for Blencathra.

If you fancied Northumberland then it makes Slaley Hall or the Gilsland Spa workable. Between us I'm sure a bit of thought could be put into that.

I don't know if @ian and julie could rustle something up for you spa/walking wise on the North York Moors or even the Dales.

Thank you everyone for the great suggestions. We decided we will stay in Grasmere at the Daffodil Hotel and Spa.

I think the Daffodil is a very good choice. It matches your 'wish list' very well.