And to add to the above - No.
The simple answer is this.
Not one Celtic historian, not one Celtic book, not one Celtic website makes the claim that they sung it first. Not a single one. It is a myth which has grown up in the last three or four years on the internet.
There's newspaper evidence, radio evidence, TV evidence of Liverpool singing it in the early 60s. No such evidence exists for Celtic. None whatsoever, not even a passing mention - which you'd think was strange if they'd sung it first.