I'm an adult SLT, my specialism is cancer and voice.
Sorry to hear about your Dad's diagnosis, and the difficulties he's having. I hope his tumour responds well to treatment.
Your Dad shouldn't have to wait long for SLT help at all - given his diagnosis he would be a high priority in most areas, especially as a paralysed vocal cord can also result in swallowing problems. You might well be able to self refer to your local Adult NHS SLT service, particularly as you know exactly what the problem is, so he wouldn't need to go via ENT (little ducks is quite right, for voice disorders, we need to know what we're treating). They may want to see evidence that someone has looked at his voice box, though. If you contact your local hospital, they should be able to put you through, or give you contact details.
If he's having chemo still, they might also not want to offer any active treatment until the outcome is known - the chemo might well shrink the tumour and relieve the pressure on the nerve, for example. Once the outcome is known, his Oncologist will almost certainly refer to ENT if his voice is still a problem, as there are various procedures that can really make a difference.
For private SLT, ASLTiPP (or something similar) are the official organisation, you can contact them via the website of the Royal College, our professional body (www.RCSLT.org.uk).
As far as amplification goes, I've recommended the waistband ones before, though I can't say if they would actually help your Dad. Often, if you only have a whisper, a loud whisper (against amplified background noise) isn't that much more audible. If he can try your husband's one, that'll give you an idea of whether it's likely to be successful.
Some people are dead against the idea, but if your Dad's au fiat with computers or mobile phones, there are some excellent, and very cheap, text to speech apps available. He would type in a message, and it reads it aloud for him.
I hope some of that is useful.