Hi Obbelix,
I believe your mystery tree is a Wych Elm, the flowers are not flowers but winged fruits which on the Wych Elm are visiable before the leaves unfurl. The bark is greyish with thick straight ribs. This is a tree that is somewhat resistant to Dutch Elm disease and is now the only common Elm in Britain and northern Europe.
What lovely pictures, BF was anxious to know where the gardens were but was a bit disappointed to find out that they're in the Netherlands.