It’s worth looking around before being referred, since there are clinics that have more resources – the one I eventually got referred to has 7 sleep docs, about 11 technicians, and runs 7 days a week. They told me they could have me in within 2 weeks if I wanted to take whatever doc was available. It took longer, partly because of summer holidays I guess, but also because the director of the clinic assigned me to a specific doc. He’s my current favourite person in the whole world, he’s worked a miracle for me. 
Also, he deemed my case urgent, and fast-tracked me to a sleep study within a week.
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