I have serious doubts that a spring-driven vehicle would have any sort of useful range.
I don’t think that usefulness is the primary goal for what would be, essentially, a sports vehicle. And the one train they built went two miles, so we know that at least that far is possible.
Yeah, but that’s on a level track not open ground or dirt/cobblestone roads. The ones shown had winding machines they had to link up to for rewinding. That’s fine when you have a large supportive infrastructure in the areas you want to operate in, but without that you have to stop and wind the sucker by hand, which would take quite a while.
Not if you had a wizard do it! 
Even if you accept the need for cast steel, that’s not a particularly high hurdle to clear, given that Wootz steel was being produced in 300AD and the Chinese were also doing something similar in that timescale by melting together cast and wrought iron. So if someone was extremely forward-looking and immensely stupendously wealthy there’s probably no reason why they couldn’t have a steam car or an IC car, maybe based on something like a Hot Bulb Engine for the sake of simplicity and easy fueling.
Bear in mind that one advantage of steam over IC is that you have torque from zero revs, and so don’t necessarily need a gearbox. I was thinking a steam turbine might be simpler than a piston engine, or if there’s too much precision needed there, something based on the concept of a Pelton waterwheel.
They charge to damn much nowadays. Fifty gold and a live goat just for a lousy consultation! It’s even worse with the big firms like Merlin-Elric & Snape.
It was also pulling a passenger car when it went 2 miles. If you formed it into a car, it would probably have something like 1/4th to 1/5th the weight to pull. And if you add a gearbox of some sort, you can extract more power from the spring to make it up hills, and with intelligent braking clockwork, you can rewind on downhills.
You’d be fairly well guaranteed a specific range that you could go.
As a luxury vehicle to drive around town just to show off, that’s not really an issue.
And the important thing is that outside of manufacturing the springs, even the ancient greeks could have built one since it’s all just gearwork, and if you did set up roadways with regular stops that had pre-bundled coils, you could actually get a fairly good transport system going.
The building of the tools to make the tools to make the tools is what will kill you, on top of which, unless you’re very familiar with machining and fabrication, you’ll have to rediscover a lot of technology on your own. When they were first building steam enginers, it was a point for bragging that they could build cylinders that fit into apertures with a tolerance equal to “the thickness of an old shilling”. thousandths of an inch that ain’t, with resulting loss of pressure, and a lot of wear and tear on the parts. The idea of forging big hunks of metal, and coming up with drills hard enough to bite without wearing, and then actually using thse drills to make something is pretty daunting, especially if you haven’t got any sort of base technology to fall back on.
Read L. Sprague de Camp’s Lest Darkness Fall or Larry Niven and David Gerrold’s The Flying Sorcerors for well-imagined attempts to build high tech in a technology’less culture. de Camp’s Martin Padway, who never gets the hang of making gunpowder, and can’t get enough interest by the locals in building and maintaining a telegraph somehow seems more likely than Mark Twain’s Hank Morgan, getting entire factories going in Arthurian England.
Harry Harrison decribes an automobile built on a low-tecj planet in the third of his Deathworld novels, but I think there was a low tech base to work with there. And the auto was definiytely steam-powered.
A bump, becuase I found this book which gives an account of a spring car built in 1644. It also discusses other propulsion methods from that time.
Do you have catapults? if you do you can basically change it to pull a string around the back or front wheeles. As far as winding rember if you have draw bridges etc you acn use gears or pullys or comealongs. Attached to the non drive wheele it would wind it up by rotating the “drive shaft” in the oppsit way. When winding is finished the handle is removed. Then you hop in your “car” release a mechanical break and off youd go. One down fall is no reverse. For stering check out a childs wagon for an idea. The goat bladder tire is not a bad idea but its extreamly weak. I’d use old swords and animal hides for traction the “tires” would have to be wide.
Jist rember Your the “dm” or whatever. It may look stupid and sound as fake as a flying bull. But they made it its not really earth any way
If Bulls do fly are they red lol