I think it is dangerous to use a statute to change sporting seasons. In the current political climate its only going to go one way. Far far better to regulate by voluntary measures. I dont know of any woodcock shoots that shoot in October, or November for that matter. Most pheasant shoots don't allow Woodcock to be shot anyway. The idea that 160,000 woodcock are shot in the UK is just another Packham lie. There isnt even a grain of truth in that number and he knows it but so much of his campaign of bigotry and hatred is based on lies this isnt a surprise.
Currently Woodcock are a globally very common bird but sadly the UK breeding population is in decline, but this is in no way connected to shooting. After the start of December the vast majority of Woodcock seen in the UK are migrants and so for the last few seasons all the major shooting and conservation organisations have recommended no woodcock shooting before December. This seems to have been pretty much universally accepted in principle. The reasons Packham is campaigning for a change in season are largely about his own desperate need for self aggrandising publicity and his ongoing hatred of normal people who enjoy hunting and fishing.
So dug around a bit (as I have yet to try game shooting, so no prior knowledge or experience). The 160,000 apparently comes from GWCT's "studies" on it. An article on ShootingUK would suggest that a more accurate number would be 70,000-100,000, and as you say, the majority (if not all) being shot in December. This ties in with what I've seen off Facebook, where most of the shooting respondents concur.
Indeed, some sentiment, as with Aled, that the petition doesn't really matter, as it's considered irrelevant. However, typing that all out has let me think that it's perfect fodder for further restriction on woodcock shooting. They'll push the season forward to
December (For Aled, not November - though don't know if your's was a typo

), and report that there is still 100k+ bird being shot and therefore further stops are required...