You'd barely notice a difference
... mainly because iron is so incomprehensively common.
Iron is about 1000 times as common as copper and 20,000 times as common as tin in the Earth's crust.
So, even with 10-1000 times less of it, iron/steal would still be easy to make in significant amounts. At 10-100 times less common, you would probably not notice much of a difference in our world's history. Steel would be a bit more expensive, but still a pretty universally accessible metal. 1000 times less common, and it would probably extend the period of time where Iron and Bronze were used interchangeably much longer, but not really prevent the use of steel since it is still the better option a lot of the time. Instead of iron replacing broze, you'd just see each used based on when one is better than the other, or when one is more available in your region.
At 10-100 times less common, no industry would be significantly impacted from developing, but some places in the world might be iron scarce. This could affect the balance of power between economies throughout history such that some empires may rise instead of others and from this you could speculate all sorts of butterfly effects, but by in large, there would be enough places to acquire iron in that world that no country would be fully cut off from having it as a readily available resource.
At 1000 times though, you may also start to see the industrial revolution impacted, but not really prevented. There would still be plenty enough steel for mechanized production machines to remain economically viable. So early industrialization like grain mills, textile factories, etc. would still happen. But... things like trains, automobiles, and highrises use a LOT of steel, and this is where you would see the scarcity bottle neck maybe start to affect you. Just like copper started becoming scarce when we decided to wire and pipe up our entire world with the stuff, iron might become scarce if we tried laying down too many railroad tracks, making too many car engines, or making a bunch of steel framed highrises. There would still be enough viable ore to go around to get you started, but you'd often have to go farther to find it which would make it more expensive.
The transportation revolution would seem to be in jeopardy because of this except for a very important discovery that happened in 1888 called the Bayer Process which is where we learned to mass produce aluminium. Aluminium is even more common in the crust than iron; so, the Bayer Process opened up a virtually limitless supply of metal even without abundant steel. By replacing most of our bulk steel with aluminium, the industrial revolution could stay more or less on track. Aluminum, like copper, needs to be alloyed to become comparable to steel in strength, but instead of needing something rare like tin, it normally alloys with trace amounts of magnesium, silicon, and/or zinc which were all easily isolated elements by 1888; so, by the time the automotive and high-rise structure industries really starts to take off enough for steel supplies to be a problem, we'd already have enough enough access to aluminium alloys to pretty much replace steel even at very large scales.
The bottle neck would be cost, new aluminum costs about 3x as much as steel per pound to refine but is twice as strong for its weight. This would make things like sky scrapers much more expensive at first because you would need a similar weight of aluminum as steel to hold up the weight of all that concrete, but things like automobiles would be less affected because you are only engineering to the weight of chase. Either way, the cost of aluminum would cause an initial adoption issue for a few years, but would not stay a lot more expensive for long.
Aluminum is much cheaper to recycle, form, and transport than it is to refine. So, recycled aluminum products are only about 1/2 the cost of steel; so, as your civilization starts to have enough old stock to blend with the new stock, the cost of aluminum will drop to prices that could be comparable to steel today.
All this would really mean for anyone today is that cars would be a bit more expensive maybe making public transportation a bit more common and high rises would also not be as high. But on the surface, society and our history would still be pretty recognizable.