Off the top of my head...

Tatton Park, Manchester, UK - a small perfect vignette shipped over from Japan in the 19th Century but you will need to make special arrangements to walk within it rather than view from outside.

St.Mawgan Bonsai Centre, Cornwall, UK - loose woodland stroll garden, not slavishly Japanese but has wonderful relaxed vibe. Bonsai, plants and accesories sales.

Pine Lodge Gardens, Cornwall, UK - teahouse with terrapin pond. New compared with existing English woodland walk garden but settling in nicely, some unusual plants in sales area.

Dartington Hall College, Devon UK - zen garden nestled within the college buildings, hard to find.

Probably the best two examples to view in the UK are Tatton Park as suggested above (though this was hardly shipped over from Japan in the nineteenth century)it was constructed in the early twentieth century - the lanterns, tea house and Shinto shrine originating from Japan - a very fine moss garden and also Lauriston Castle, Edinburgh- much more contemporary, the work of Takashi Sawano - as Edinburgh is well worth a visit in its own right then Lauriston should be added to any itinerary.

Holland - Clingendael, The Hague, the same early twentieth century construction as Tatton - also a very nice garden to visit.