Also - might be worth chasing the application. I am sure that you have sent your logbook using a signed for delivery, has the delivery been made?
I would strongly recommend calling them ASAP to make sure that they have received your application, and that it will be processed. 5 weeks seem abnormally long to get your licence issued. Although this may have something to do with recent changes in legislation regarding the issue of licences.
Best to call them to get an update, and maybe even a time-frame with regards to licence issue.