Jackie, I went back when it wasn't raining and I could look around and found that this plant is a substantial climber with strong woody stems up to 10 mm thick (see additional photos). It had climbed up into a young angophora and I didn't see that bit as I was pulling fireweed at the time and had my head down!
I wondered about Billardiera. The flowers look right but the leaves don't. It's in a place where I wouldn't have expected to find a planted species, but you never know. If it was planted, it's been there a good while now with no spread that I can see. I'll keep an eye on it, though. Thanks Jackie.