Wow - that's an extreme example of misrepresentation!
I don't think I have ever seen a taxi waiting at Pangbourne, and would be surprised if there have been any at Theale for many years. If you had booked one to pick you up, then you might do it from either, except that once 2 or 3 at most has booked them I think you would have exhausted the local supply.
It's a 30 minute walk from Theale Station, closer to a hour from Pangbourne. I checked Englefield House's website and the website for Runfestrun (who appear to be the organisers) and neither repeat this claim there. The latter simply says it's 5 minutes from the M4 by car - which sounds about right.
Not a bad walk from Theale Station,especially if one is aware of the permitted path through the ornate gates on the A340. About 15 months ago, the lady who then supervised the "5-a-day" market garden in Englefield put her hand in a mower whose blade hadn't stopped. Once she'd recovered (more or less) but couldn't yet drive, she conscientiously returned to work, taking the train from Bramley, changing at Reading for Theale, then walking to Englefield.
But from Pangbourne ... there's a pavement along the A340 to Tidmarsh, but then it gets very dodgy. If I had to do it, I would turn left past the church, then right into Moor Copse, scrambling up to the bridge over the M4, down the other side to Malpas and across the A340
