Hope to get some good advice on this.....

My neighbour and I share a hedge along our border. It is approx 5 feet
wide, 10 feet high and 65 feet from house to end of garden.

We are agreed that it needs to be reduced in both height (by 4 or 5 foot)
and width (by about a foot).

I don't know what the hedging comprises of except that it has lots of yellow
flowers in the spring and is not an evergreen.

So to the questions..... when is the best time to do the work and should the
height be reduced in stages?

sounds like forsythia

i have one a similar size, and have reduced it by that amount over a 3 year
period, with great success. prune directly after flowering for the best
display next year.

a friend tried pruning harder than that and killed a couple of the shrubs
off in his garden.

it should then flower better than ever although mine now grows very
vigourrously, having reduced it stages i can now take off all one years
growth in one go after flowering.

chaz