You're correct - it doesn't take a month. It took them 9 weeks exactly to process mine.
I did the online application.
7.5 weeks after applying they said I'd not sent the course completion certificate through so they couldn't progress my application. Turns out there's a different cert for online applications. Check that your course completion cert has all the hour breakdowns, exam passes etc. listed on it. If it's the traditional blue one then that's the wrong one for online!
Then, a week after sending them the different cert they said that because I started my training in 2004 then I needed to send in my logbook so they could examine it to see if I had done enough hours, and sent me a link to the pre-2012 requirements, the thing I'd been granted an exemption from.
Anyway, after I sent an email to the CEO I had a phone call back that afternoon from the head of licensing saying I didn't have to send my logbook in, they'd found the exemption and my license was all signed off and back in the post to me.
Long story short - I'd have had a lot less hassle if I'd posted it all in...
But either way, you're still looking at a 40-50 working day lead time...