I'd say Arbutus menziesii, which I think is commonly known (in America) as Pacific Madrone. But its definitely called Arbutus menziesii here...bit of a fussy beast, will only grow in acid soil and in mild conditions, may suddenly die as it matures, and will not get through a hard winter. Arbutus unedo is more commonly planted, its a little bit hardier and more reliable.

Looks like a type of myrtus (myrtle).

It's a lovely tree, whether it's Myrtle or Arbutus!

I should have said, Arbutus menziesii does produce fruits which turn black - Arbutus unedo has red fruits which don't go black.

It is weird that it appears to have opposite foliage, which is not the norm for Arbutus species.

I'm finding that impossible to tell from the pic, Tugbrethil...

Mainly from the watersprout shown in the second picture, more subtly visible in the branching pattern, and in the foliage around the fruit--I concentrated on where the leaves are attached to the stems.

You're right, Tug, the leaves on the watersprout are opposite - in which case, it might be Luma apiculata, a myrtle variety (originally suggested by Longleaf above):-

http://plantlust.com/plants/luma-apiculata/

but it would be very surprising if it was growing so well in Chichester - normally not guaranteed hardy and only survives down in the West Country here, and even then keels over in a cold winter. The other possibility is its rarer relative, Eugenia candolleana:

http://www.wikiwand.com/en/Eugenia_candolleana

http://www.daleysfruit.com.au/Eugenia-candolleana-Rainforest-Plum.htm

but again, not entirely hardy here. Links are for Melew to investigate - the second one gives a description of the fruits and what they look like when cut open, which might be a useful thing to do, Melew, see what's inside them.