Ironbridge put up the wrought iron works in the expectation that there would be a small but steady ongoing market that would approximately cover the cost of fuel for the furnaces, enabling periodic demonstration of both puddling and rolling. However, steels began to be produced that were suitable for blacksmiths to hammer-weld, reducing the market for genuine wrought iron, and fuel costs became ever more expensive. So Blists Hill rarely can afford to fire up the puddling furnaces and limit demonstrations to re-rolling large thick pieces of wrought iron scrap into smaller cross section bars as this only requires the reheating / mill furnaces to be heated up. The main source of large cross section wrought iron scrap in recent decades has been enormous chain links from naval dockyards. These are cut into pieces, heated, and rolled down to small cross-section merchant iron.
Having used the re-rolled stuff myself, I found the quality a bit variable, came to the conclusion that you probably can't match new puddled iron.
As far as I know Ironbridge rolling mill hasn't been used for a while now (5 years?), the firm that was having re-rolling work done there has their own smaller mill so I guess there has been no call for large sections lately. The re-heating furnaces are oil fired so getting them going wont be cheap, then theres the engine running the mill and the hammer...
The puddling furnace last ran in the 90's I think, when I visited in 03 I had a chat with one of the guys who organised it and apparently the furnace has to be rebuilt each time, there simply hasn't been funds for this and given that it was maybe 20 years ago now I wonder if the people who know how the process is carried out because they did it for a living are still around...