Upvote:2
It seems difficult, for two reasons:
Note that if they decline to process your application because you are not a resident, you should in principle be informed quickly and get a refund so there is not much to loose in trying. But if they do process it and just happen to take 10 days, you won't know it before it's too late for you and you will have paid the fee for nothing.
Unfortunately for you, the Italian consulate in London requires applicants to go through a “Visa application center” outsourced to VFS. They tend to add delays more than anything and will probably start by complaining about the fact you are not a resident (they do list it as a requirement) and the unusual purpose of the trip instead of forwarding the application to the consulate so it makes it even less likely for you to manage to get a visa in a week.