According to the Lagan Canal Trust,
The Lagan Navigation also forms part of a wider all Ireland waterway network. This network of waterways once traversed through the towns and cities of Ireland delivering goods and produce, helping to shape the economic fortunes of the country.
I would be grateful for information about any goods or produce that were ever carried from the Shannon, or from the Royal or Grand Canals or the River Barrow via the Shannon, through the Junction Canal in the Ballinamore & Ballyconnell Drainage District [later called the Ballinamore & Ballyconnell Canal and later still the Shannon–Erne Waterway] and then the Ulster Canal to Lough Neagh or any of the waterways connected therewith. Or, of course, in the opposite direction.
As far as I can tell, outside the sales blurbs written by engineers seeking employment and waterway owners seeking subsidies, there was never a connected all-Ireland waterways network; nor was there ever any need or demand for such a thing.
Any more than there is now.
Irish waterways history 
Well why did the Irish build 3 canals that all went to Lough Neagh, then? The Lagan navigation could’ve gone from Belfast to some metropolis like the Cathedral City of Dromore, the market town of Ballynahinch… but no, they dug it to Lough Neagh. Why? If not to continue water-bourne transport of goods across the Lough and onwards elsewhere? Likewise, my all means dig a canal to the port of Newry, but why not start it at somewhere populated like Armagh or, ahem, Lurgan? But no, it starts at Lough Neagh. Why?
Coal. For Dublin. bjg
Does transhipment of coal from thence to Dublin not count as an all-Ireland waterways network then?
The sea is not usually counted as an inland waterway. And the existence of the sea meant that communication by water between Dublin and Newry, Belfast and other places was available, offering higher speeds, larger vessels and less cost than any canal could offer. bjg