This is what I found from HO website. It clearly says existing work permit holders can start working as long as the new work permit is approved. (The last paragraph).
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When can I start work?
You are unable to start work until you have received approval for your leave to remain application, unless you already have leave to enter or leave to remain in another immigration category that allows you to continue or commence full time work.
However, if you are applying to extend your leave in your existing category you may continue to work whilst your application is being considered, provided:
* You have previously been granted leave to enter or remain in the UK in a category that permits you to work
* You submit a valid in time application to extend your leave to remain before your existing leave expires
Examples of these categories include existing HSMP applicants. If you are in any doubt, you should phone the Border and Immigration Agency on 0870 606 7766 or visit our website at
www.bia.homeoffice.gov.uk before taking up employment.
Also, arrangements have been put in place whereby existing work permit holders can start a new job on the strength of the work permit permission letter, providing their existing leave to remain covers all or part of the period of the new employment. Individuals will then have to submit a further leave to remain application before their existing leave expires, or within 6 months, whichever is the sooner. If the individual's subsequent further leave to remain application is refused, the individual must cease the new employment and Border and Immigration Agency will take action to curtail the leave granted in relation to the previous employment.