I've just contacted the UKBA and they said that you can continue to work under the conditions of the visa you have while your new visa is being processed.
The information related to this can be found at the bottom of this page:
http://www.ukba.homeoffice.gov.uk/visas ... ding-stay/
where it says:
If you make an application before your authorised stay ends, your existing immigration status will continue until your application is decided, even if the decision is not made until after the end of your permitted stay. If your existing visa or other permission to stay here allows you to work, you can continue to do so until your case is decided.