Unhappy plum tree

The tree has two trunks to it which grow from a split at the base. One main trunk has lots of leaves, where as the secondary trunk is bare of all leaf growth, but does seem to have some new shoots coming off it

This isn't a graft by chance?  The "secondary" trunk might be the root stock that may be trying to grow - sometimes something (lack of water, too much water, heatwave, cold winter ...) can encourage the grafted root stock to send out suckers and maybe this is something similar?  Even if it is not the graft, having 2 trunks might be diverting water uptake in times of water stress so that some branches do not get enough (and hence, die-back on some leaves)

In the photos I saw lots of trees in the back ground as well as stuff underneath the tree - plums need good air movement through its branches otherwise the humid conditions can encourage problems.  Maybe prune it midsummer to open up the space between branches in the centre (always prune plum trees in summer when it is dry, not in autumn/winter as it increases risk of silver leaf disease) and nip off those dead branches.  You might want to wipe the shears with a disinfectant in case of disease - you don't want to transfer it to good wood, and burn any trimmings from the tree.

With regards fruit - wasps and other pests like the plum moth larvae can invade fruit and this causes them to show problems.  If fungal, sometimes cutting back the dead and damaged branches may reduce infection - never heard of it before, but I saw something called pocket plum.  Also, to ensure the tree remains productive, you should thin out the number of fruit on the tree (wait until after the June drop to get a better idea - this is usually when non-fertile seeds initiate fruit to drop off trees so that the others can develop normally)  Also look for ants and aphids.

EDIT:  maybe put some potash as a top dressing round the base - this can help trees against fungal problems.

Here's some more pictures I managed to get last night.  I've been reading about silver leaf disease and think I could be a victim.  There is a brown centre to a branch I broke off and the pictures show the colour of the tree, i.e. very silvery next to the green of the bay below and the trees in the background.  Is there anyway to truly confirm if the tree has the disease?

Try contacting one of the orchard groups http://www.orchardnetwork.org.uk/content/south-east I am sure someone will point you in the right direction to confirm if it is.