I can't imagine what the neighbour has to be peed off about.
Their roots are encroaching on your property. They should be doing their best to accommodate you... not the other way around.
Ubasses link makes a good point that you "may" be liable if the tree suffers. In the real world, just imagine if those roots had made it to your house foundations, possibly causing "heave" and affecting your structural stability.
Perhaps, to be practical, it may be worth getting opinion from a tree expert. I know it's all extra expense, but I would rather get this on a clear legal footing now rather than later.
I've dealt with roots several times, but not with this aggravation. Usually, cutting the roots and installing a barrier has no particular effect on a healthy tree.
I say barrier, not for your service trench, but for the foundations of your proposed extension.
Root barriers are often simply mass concrete fill in the foundation trench together with a compressible "clayboard" (nothing more than a fancy polystyrene board a couple of inches thick.) When you have your trenches inspected by Building control, they should advise.
Just bear this in mind in case you do come across the roots in your extension excavations.
Get yourself armed with some professional advice, Beds. It's well worth it.