Its OK to use fossil fuels as chemical feedstock in manufacturing. As long as we don't literally light it on fire for energy we can deal with the resulting reaction gasses.
In 500 years the idea that we ever burned our most valuable manufacturing chemical to keep warm is going to seem crazy. Petrochemicals are incredibly useful for making things.
Cement needs lots of heat to make it - from coal too. Also, cement is made from CaCO3 (limestone, the shell of ancient microorganisms). It releases the CO2 it contains when transformed into cement.
Carbon in form of coal is currently used for three purposes in steel production:
1) Heat up the ore to high temperatures
2) Reduce iron oxide to iron.
3) Steel is an alloy of iron and carbon.
Only for the third of these carbon is essential, and that requires some tens of kilos carbon per ton of steel as opposed to more than 2 tons carbon per ton of steel. The two first ones can be replaced by electrical heating and hydrogen respectively. There are currently being built some factories in northern Sweden for doing this, using hydrogen produced by hydropower. Without sufficient tax on carbon or customers willing to pay the extra for "green steel", it is not cost competitive for now.
The coal used for reduction of iron ore to iron can be replaced with hydrogen through direct reduction. See Hybrit which has working industrial scale demonstration plant today, though at reduced capacity. Full capacity plants are planned in multiple locations by 2036.
My guess is that the coal could come from wood? The heat could definitely come from nuclear power.
However, the processing of the limestone might be more difficult. But then again, that also seems like insignificant emissions when the other ones are taken out, no?
Making cement without baking the limestone, which itself releases carbon plus the burning of gas to make the heat, is possible and is being piloted now. We just lack the will to mandate these changes.
In 500 years the idea that we ever burned our most valuable manufacturing chemical to keep warm is going to seem crazy. Petrochemicals are incredibly useful for making things.