Jon,
I doubt very much that the problem is in the transmission itself. Over the years there have been many reports of the car going in to ‘limp home mode’ and I do not recall any being due to a faulty transmission unit.To get a flavour of the sort of problems encountered enter ‘Limp Mode XJ40’ in the archive search engine facility. There are pages of posts on the subject - this being typical -
A favourite suspect after the firewall relays seems to be an oil soaked TPS.
I believe Bob Gauf’s post answers your question about a code reader for these cars.