Well, I'm admitting defeat on the 1861 having spent most of the evening searching. But, I do have some info whihc might help you go further back. The couple didn't seem over-fond of registering the births of their children apart from Sarah Ann Weir registered March quarter 1860 Stoke Damerel (covered Devonport) vol 5b page 318. The birth cert would give Mary Ann's maiden name.
Having given up on 1861, I had a quick look at 1851 and found
39 Finch Street, Liverpool HO107/2184 folio 244 pg 7
Thomas Weir head mar 30 soldier Royal Artillery b. Ireland
Mary Ann wife 18 b. Ireland
Ann sister unm 25 b. Ireland
John son 1 month b. Lancashire, Liverpool
That's the same birth years as 1881, he's in the army, same birth places? It looks a possible. His sister has come to stay to help with new baby by the looks of it.
Also spotted this marriage registration June quarter 1850 West Derby(Liverpool) vol 20 page 1041, on the same page Thomas Weir and Mary Ann Hullahan. This is when there are 4 couples to a page, but if you get Sarah A's birth certificate, it would confirm maiden name - unless you already know it of course!
I don't know what to suggest for 1861 - perhaps they're visiting family in Ireland? On a missing sheet, not enumerated or I just can't see them. It might be worth posting on the army board? If Thomas was in the Royal Artillery, there will be records at the National Archives, but I have no experience of finding these and I don't know if the records would tell you where he was at a particular time. I would think you could trace this regiment though. Again a query for the army board.
hope this helps.
modified to add forgotten census ref