If the oil cap is left off, the oil won’t all drain out at once. It’s not like leaving the oil drain plug off. If that happens the oil will be all gone within a few minutes.
The oil cap is in the valve cover, under which is the camshaft(s). These shafts are being constantly sprayed with oil when the engine is running. If the oil cap is missing, there’s a hole in the cover and some of the oil will spray out that hole, and eventually most of the oil will be sprayed out that hole if it goes on long enough. But I’m not sure how long that would take. Quite a few hours of driving I’d expect.
Since you aren’t a DIY’er I presume, don’t fix your own car or change your own oil, you are not used to dealing with these kinds of problem in the engine compartment, you well might not notice anything wrong. You are within your rights to depend on the oil change shop to make sure everything is put back together properly after an oil change.
In the future I’d recommend – if you are inclined – after oil changes, before leaving the shop to turn on the engine, pop the hood, and look for oil leaks coming from the top of the engine through the oil cap area, and look under the car to see if there are any oil leaks down there. Then drive the car 5 miles, and do the same thing. And again one more time the next morning, including checking the dipstick.
The aftermarket oil cap remains a mystery though. Any ideas about that?