Doug,
Here is a list of the people called J Slattery who served in the Boer War:
A J Slattery New South Wales Lancers
J Slattery 1st Battalion R Munster Fusiliers
J Slattery 4th Yorkshire Regiment
J Slattery 9th Kings Royal Rifles
J Slattery 27th Company A S Corps
J Slattery 2nd Brabant's Horse
J Slattery 6th Company R G A West N Doy '2
J Slattery 1st Imperial Light Horse
J Slattery 2nd Battalion Manchester
J Slattery Imperial Light Infantry
J Slattery 4th Battalion West Yorkshire
J Slattery 3rd Battalion The East Lancashire
J Slattery 3rd Battalion Kings Royal Rifles
J Slattery 1st Battalion The Royal Irish Regiment
J Slattery 2nd Battalion Manchester Regimant
J Slattery Harbour Officials & Others At Cape Colonial Ports
J Slattery 4th Battalion Yorkshire Regiment 3rd Battalion Yorkshire
J J Slattery 2nd Battalion Australian Commonwealth Horse
J A Slattery Kaffrarian Rifles
J E Slattery 5th Battalion Royal Fusiliers
J P Slattery 1st Battalion Port Elizabeth Town Guard
J P Slattery 1st Battalion King's Own Scottish Borderers
J W Slattery De Lisles South Australian (5th & gth I B C)
James Slattery 1st Imperial Light Horse
John Slattery 6th Company Western Division Royal Garrison Artillery
The two probably Irish men are:
1670 Pte J Slattery, 1st Battalion R Munster Fusiliers. He was not involved in the relief of Ladysmith
6061 Sgt J Slattery, 1st Battalion The Royal Irish Regiment. I cannot find his QSA record but the RIR were not involved in the relief of Ladysmith either.
Some of the other men could also have been born in Ireland.
To narrow the search, it would be necessary to cross reference the units with involvement in the relief and then to drill down into each man to see if he received the clasp and to check whether there are any existing service papers.
If you have a subscription to Ancestry, the first part of this would be possible. A subscription to FindMyPast would be needed to check online for service papers.
Best wishes
David