London to Stourhead National Trust

From the Traveline website you take a train to Gillingham (a London Waterloo to Exeter train, not the better known Gillingham in Kent), then catch a bus number 58 (see attached timetable) to a village called Zeal, then a 1 1/4 mile walk!
https://images.southwestcoaches.co.uk/images/pdfs/58gillingham-zeals-mere-gillingham.pdf

Good luck.

My goodness. The Stourhead website isn't very helpful. I was surprised, because often the "directions" for a venue will include tips about taxi services, etc., and a map. This one is map-less and almost cryptic:

By train: Gillingham 6½ miles; Bruton 7 miles

By bus: from Shaftesbury, alight Zeals, 1¼ miles

By road: Please arrive at Stourhead via the main road network,
following brown signs off A303 or the B3092 from Frome. The main
carpark postcode is BA12 6QD.

Cycling: Wiltshire Cycle Way [with a link to the detalis]

You could contact them directly to ask about taxi services from the train stations. There's an email address on their website.

From Zeals, you need to walk up Bells Lane. Find route on Google Map and go to street view and follow the road you will walk.

I have only been to Stourhead by rental car, but I want to suggest that taxis be reserved and necessarily, prepaid. We were unable to change our taxi return to rail from a (different) garden we finished early, because some taxis are contracted for school transportation hours.

I've got the paper map out, the 1 1/4 mile walk (actually far closer to 1 1/2 miles) isn't actually on a safe road, there is a different and much better route from Zeals which cuts round the border of the estate, that is about two miles each way- that is still on a minor road, there is NO footpath route from Zeals.
The footpath access is from the next village- called Mere- about 2 1/2 miles with a climb up the well named Long Hill, then a reasonably steep descent down the other side
I'd be happy to do it personally (as a round trip), but am aware that others wouldn't be and it's not exactly looking like a route to recommend.
The bus information given on their website is also wrong as Shaftesbury does NOT have a railway station, nor does the direct bus go from there. If you were in Shaftesbury (having got there by other means) you would have to change bus at Gillingham anyway.
The cheapest fares from London are £11.80 each way (good availability) but I would buy what is called a Semi-Flex Return for £42.50. This is valid outwards on the specified train, but return on any train- so you aren't fretting about time.
The train takes two hours each way.
Gillingham is a fair sized town but the station information on South Western Trains says there is no taxi rank, so you would have to pre-order one.
The other note of caution is that at the moment there are not many through trains from Waterloo due to a landslide near Basingstoke 3 weeks ago. The latest date for full re-opening is 22 February, and it is unlikely the date would move a further two months. But until the line really is back open it is best to be cautious.

There’s a direct train from London to Bruton. Bruton is a lovely village with a contemporary art gallery and many independent shops and restaurants. It’s quite set up for visitors so I’m sure you could get a taxi to Stourhead without too many problems.

Thank you all for your replies! Very helpful.

I agree with Tim re reserving a taxi in advance. Some years ago (before I began renting a car when in the UK), I rode the train to Gillingham then rode a taxi to Stourhead. I did not have an advance taxi reservation and was able to get a cab at the Gillingham station. However, the taxi driver told me that he was already committed to "school runs" that afternoon. I was able to time when I finished at Stourhead around the school schedule so that I could get a cab back to the Gillingham station. But, I learned a valuable lesson and decided that in future similar situations, I would reserve a cab in advance. AND I agree that Stourhead is SO worth seeing. I've been 2 or 3 times.

For the sake of completeness, as the National Trust travel information was copied to this forum.
If you wanted to get to Shaftesbury by bus to then go to Stourhead, there is a very thin service from Salisbury (which would be a nonsense route as there is an hourly train from Salisbury to Gillingham), or the only other route is from Poole (on the south coast) - every 2 hours, run by Damory Coaches with a change at Blandford Forum- a route which takes nearly 2 hours.
Poole and nearby Bournemouth are big seaside resorts, and well served by rail from London.
So that is the answer to the cryptic clue. It takes 4 buses to get there that way, not 1 as claimed.
There used to be a book in the UK called the Great Britain Bus Timetable- listing every local bus service in England, Scotland and Wales. That provided the answer to the National Trust's cryptic clue, after converting those timetables into current ones. You never could get that book on line but it was and still is the 'Bible' to UK bus routes. For all that it is out of date it is far better than any on line planner.