It's a difficult situation for you Mark, if not impossible but I never like to use such words or Can't.
What you have working against you is:
. as you have recognised yourself - the lack of formal qualifications - and sounds like you've had a good read of immigration options.
and then with sponsorship:
. there's still a requirement for sponsored people to be qualified.
. most tiling done here within Australia is done by smaller sub-contractors, themselves ironically not necessarily being qualified but having skilled up through experience.
I'd not expect that too many of them if any have ever considered getting involved with immigration on sponsorship.
The government regularly reviews immigration requirements, and we have even had transport industry lobbyists wanting immigration to provide for truck drivers - asia being seen as a source to some horror! from those who have experience of roads in some asian locations but to be honest I would not be holding my breath on something occurring in regard to your own skills.
If you have Australia on your scope, I'd have a good look at the SOL and get an idea of what skill you could easily get qualified in (age permitting) for instance, something akin to your trade or area of work of you or your wife's if she works, say real estate sales for instance?, hairdressing? and get a year or so of experience and then go from there.
Could be better than attempting to batter down a brick wall.
Just a quick google on Tiling Companies brings up a myriad of hits -
Tiling Companies - Google Search
and you'll have to sift manufactureres and other hits from actual companies, and there's no harm in having more than one poker in the fire, so no harm lost in emailing a few companies - outline your experience/portfolio in dot point form.
And you never know your luck!, as to getting a response, but you'll still have the qualification issue - any chance that you have the equivalent of our TAFE (Technical and Further Education) courses here that could see qualifications being fast tracked because of your experience.
But maybe it's something you can work on while you look at other qualification options.