Plainly had the Cotton Gin's invention been delayed it would have had consequences on Society and Economy in the Southern United States.
And on the north, too. There's a very good case to be made that without the cotton gin, Indiana and Illinois would have turned into slave states. They came surprisingly close in OTL even with the Northwest Ordinance.
Without the cotton gin sucking slave labour south, the existing slaveowners in southern Indiana and Illinois would probably have been there in greater numbers. Slavery would likely become entrenched north of the River Ohio, with rather unpleasant long-term consequences.
Gavin Wright has published some interesting papers on this; one of these days, I need to get around to writing up a summary of some of them. But the short version is that slavery was profitable in far more crops and regions than you might suppose.
Would it have had effects on industrialisation in England and New England?
Absolutely massive effects. Cotton was by far the largest part of the textile industry, and the textile industry was an immense portion of the industrial revolution. It wasn't
all of the industrial revolution, of course, but cotton textiles and the innovations spawned from them were an immensely important driving force.
Not all of these ways were obvious, too. For centuries (if not longer) China and India (especially China) had been running at a net trade surplus with Europe. China and India had lots of goods which Europeans wanted and would pay premium prices for, but there weren't really that many goods which were traded the other way. The difference was made up with bullion, mostly silver and gold from the New World.
Cotton textiles exported from Britain (and elsewhere) were the trade good which finally broke that trend. They drove a lot of British profits, and helped to finance imperialism. Remove cotton textiles, and the trade sink remains around for a lot longer.
Would it also have had indirect effects, say making Irish LInnen more important for longer?
Almost certainly. The revival of the Irish linen industry in OTL can be traced to the ACW and the lack of supply of cotton. Absent a cotton boom in the first place, various kinds of linen would have maintained more importance.
Other fibres did exist, of course. Long-staple cotton could be grown profitably enough and used in textiles, but it's a very finicky plant in where it can be grown, and even then the yields per acre are lower than with short-staple cotton.
Wool could be used for some textiles, too. Of course, finding the grazing lands for a few million more sheep in 1800 would be rather a problem, to say nothing of the ecological effects.
Flax/linen could be grown, too, as mentioned.
The thing about all of this, though, is that while textiles could be made from those fibres, they didn't have anything like the massive market which emerged for cotton. Cotton textiles became the preferred fibre of the emerging middle class, and changed social habits too - cotton clothes and sheets could be washed much more easily, for instance.
In short, a delayed cotton gin would have far-reaching consequences for industrialisation.