I am an Indian citizen and want to go into Hong Kong during my 15-hour layover. I applied with the pre-arrival registration form and was denied. I have sent the visa documents via fax and also separately by airmail to the Hong Kong Immigration department. At this point, I have 5 weeks before the time of my travel. Will I be able to get a transit visa within that time and, if not, is there any other way to get it quickly?
Summary: Provided your documents/information provided are in order and managed to get to the right team in time, five weeks might just be enough with some luck. Anything unforeseen happening will mean your application will miss your five-week target.
The Hong Kong Immigration Department website stated the following on the processing time for visit/transit visas (emphasise mine):
It normally takes four weeks to process a visit/transit visa/entry permit application upon receipt of all the required documents. The Immigration Department will not be able to start processing the application unless all the required documents and information have been received. Please refrain from making enquiries about the progress of the application unless it is absolutely necessary, as it may delay the processing of the application.
I have bolded the "upon receipt of all the required documents" to emphasise that this suggested time-frame only applies if they are satisfied the documents you've submitted enables them to do their work. They won't start the four-week clock if e.g. your application have a supporting document missing.
Moreover, the four weeks refers to the time between the documents arriving at their desk and them making a decision. It does not include the time taken for your documents to get to them (i.e. the right team), as well as the time for their decision to reach you.