Charlie, taking that one fairly clear pic of the fronds and assuming
all others are typical, it is Phoenix rupicola AKA 'Cliff Date Palm',
It is a native of the Indian Himalayan regions where it eventually
grows to 8 - 10m on cliffs and rocky outcrops in typically sharply
drained soil that is regularly drenched by heavy rainfall. The single
plane arrangement of the leaflets is unique amongst Phoenix and lend
the palm a far more elegant and exotic look than other dates.

It is one of the more tender species and can take quite a few years to
develop its slender trunk especially in cooler climates. I don't
think there is a mature specimen growing out of doors in the UK and I
doubt that it could survive a typical UK winter except in the mildest
of localities. I have one here that is one of my slowest growing
palms. It needs warm humid summers and mild, almost frost free winters
to grow at a reasonable rate. The winter bit isn't a problem in this
garden, but providing enough 30C+ days in summer is.