I would hate something like this happening. I don't think your neighbour is on the right side of the law by doing what he did. Here you can find the details: basc.org.uk/.../basc-air-rifle-code-of-practice. If you check under Live quarry shooting, which is what your neighbour was doing, it is clear that
(a) He needs to have permission to use the air rifle in a neighbour's land (does he? Can you check with your neighbour?)
(b) If there were a mixture of birds on the tree then do all these species fall within the permitted species for hunting with an air rifle? Even if the do he needs to have "the landowner’s permission" and do it (ie shoot the birds) "for one of the reasons allowed by the licence." Does he?
(c) he surely does not comply with the reasons for shooting:
"These reasons include:
I would follow Bob's advice and write to the commissioner, stating all the issues as indicated above (in the law) and requesting a response in writing. Add to the above that you are a nature lover and you felt personally very upset by seeing hunting from your back window as well as by the fact that you are 'forced' to hear gunshots, you feel like living in a warzone, you are concerned about what he will be shooting next as, admittedly, he has so little respect for the law as to use his riffle in an inhabited area etc etc.
My Gallery
"Any glimpse into the life of an animal quickens our own and makes it so much the larger and better in every way." John Muir