Hello, I have McLeavey's in my maternal line. Joseph McLeavey, born 1824 in/near Banbridge County Down, Ireland, joined the 65th Irish division of the British Army when he was 18 years old and in 1846 he went with his regiment to New Zealand. On his discharge he was given a piece of land at Makara (near Wellington) New Zealand, where he settled following his marriage to Sarah Ann Stoodley in 1854. I have a transcript of a letter he wrote to his brother (no name or address unfortunately) in 1874 - the only other Mcleavey he mentions in the letter is 'Peter McLeavey.'The letter talks about life in the colony and the possibility of arranging passage from Ireland or the UK.
The name is uncommon - it would be quite possible that your McLeavey's originated in County Down also - mine were Roman Catholics.