Have a look at Powys Castle Gardens when in Ellesmere, only about 20 miles from there.
Then further afield there is the National Collection of Asters at Avondale Nurseries near Coventry.

Hodnet Hall Garden in Shropshire has a wonderful garden...12 miles north of Shrewsbury on the A53, It's lovely walking round if the weather permits.

Forgot about that, lovely place. Chirk Castle is nice but not really a major garden.

I think Bressingham is in Norfolk and you could combine steam/fairground with large gardens, and see the work of Alan Bloom who bred many of the plants we have in our gardens.

Royal Botanic Gardens, Edinburgh are excellent at any time of year. Free entry too.

thanks all for these suggestions x nadia (mjdlra)

Alnwick Gardens are well worth a visit, it does cost money, but is wroth it.

http://www.alnwickgarden.com/

Also near Edinburgh is Hopetoun house. There is a very good garden centre which is large and has some difficult to get plants.

http://www.hopetoun.co.uk/visit-the-house-and-grounds.html

I put arboretums to visit into google and got a whole list, I went to Westonbirt once and the autumn colours were stunning, I completly fell in love with the Liquidambar trees........simply beautiful

I was going to suggest the Royal Botanics in Edinburgh but see Bulbaholic beat me too it. Although it's free - it does cost to visit the tropical greenhouses. Very central Edinburgh so easy to get too using public transport

East Ruston Old Vicarage in Norfolk - a plant lover's dream (and there's plant sales if you want a souvenir). You can take plants to other countries within the EU without a problem