The 'Industrial Revolution' preceded widespread coal and steel usage
According to Wikipedia,
The Industrial Revolution was the transition to new manufacturing
processes in the period from about 1760 to sometime between 1820 and
Coal production didn't really kick off until the 1860s or later. I can't find a suitable time graph for the United Kingdom, but for the US here is a chart of energy consumption over time.
Notice that coal production is close to zero until 1840 or later. However, in the US notice that Robert Fulton's steamboat made its first trip in 1807, and the first common carrier, steam driven railroad opened in 1830 (thought it wasn't very long).
As for steel, here is chart of UK steel production by year.
As you can see the chart goes back only to 1870.
In conclusion, the industries that required heavy mining output developed AFTER the Industrial Revolution.
What did happen during the industrial revolution?
To sum up the 'Important techonlogical developments' bullets from Wikipedia, here are the things that did happen during the industrial revolution, most of which would be unaffected by the lack of accessible minerals.
Textile manufacture. Important inventions were made early, included the flying shuttle (1733), spinning jenny (1764), and spinning frame (1769). Factories (as opposed to the medieval putting out system) started in 1733 (donkey powered), water power applications started at least by the 1760s. On the raw materials end, Eli Whitney's cotton gin was patented in 1794. Did not require mining.
Metallurgy. Use of coke, distilled from bituminous coal, to power blast furnaces (1709). Requires mining, although we saw from the charts that not so much was made until the 1850s and later. Charcoal and limited coal availability would be able to meet demand up until that point.
Steam Power. Newcomen steam engines from 1712, Watt's steam engine in 1778. Does not require mining, although coal made them more efficient.
Machine Tools. Boring, planing, and milling machines. Maudslay's metal lathe in 1800; Bramah's planing machine in 1802. Both of these were initially operated by human power, with a treadle. All of these can be driven by waterpower (or windpower). Alternately, steam engines driven by charcoal can be used. Does not require mining.
Chemicals. Sulfuric acid by the lead chamber process (1746), sodium carbonate (1791), powdered bleach (1800). Some ingredients like limestone must be mined, though most in relatively small quantities, others like soda ash and potash can either be mined or obtained from brunt plants. Does not require large scale mining.
Cement. Invented in 1824, requires clay and limestone. Needs sand and gravel to make concrete. Does require mining, but none of these ingredients is particularly rare. Should be readily available.
Gas Lighting. Derived from coal gas. Did required mining.
Glass Making. Cylinder blown sheet class, developed before the 1830s. Can be done with charcoal, does not require mining. Volume of glass created is not so large that it would stress charcoal supplies.
Paper making. Invented in 1798. Can be water powered, does not require mining.
Agriculture. Improved iron plow parts were made available by machine tool technology. Early inventions were machines (like the seed drill, or threshing machine) that were either hand or animal powered. Does not require mining.
Mining. Ummm...does require mining.
Transporation. Large scale application of railroads would require mining for coal and steel. However, railroad's vanquished competitor, the canal, did not. Canals developed slightly before rail, but lost to them because rails could climb hills that canals could not. Were coal and steel not available, canals are just as effective if not more so than 19th century railroads over flat land. This just serves to concentrate industry on plains.
While development of large scale industry after 1860 would be severely limited by access to coal and steel, many of the developments that made up the industrial revolution would be possible without it. Textiles, canals, chemistry, machine tools, and especially improved agriculture would provide much of the needed developments.
It is important to notice that all the required developments for later electric technologies are there. Concrete, waterwheels, and machine tools are all that is needed to build hydroelectric dams. Once electricity is properly investigated, this post-apocalyptic civilization could proceed directly from the Industrial Revolution to electric powered society, skipping over the heavy iron and steel usage.