RootsChat.Com

Trummery is right on the Antrim Down county border per the Townlands link.
The 1901 census (Drumsill, Magheragall) and the 1911 both say he was born Co. Antrim, his wife was born Co. Down and all his children were born Co. Antrim.

He married Lisburn District in Drumbo Presbyterian, Co. Down (likely his wife's church) https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1861/09613/5486254.pdf
He was aged 20 and a Farmer and his Father was a Farmer

His residence at marriage was Island Kelly otherwise Stoneyford Civil Parish of Derriaghy https://www.townlands.ie/antrim/massereene-upper/derriaghy/island-folly/island-kelly/

His father John (Farmer, widower) died Island Kelly 1875 aged 78 and John junior registered the death
http://www.rootschat.com/links/01t4g/ though no relationship is stated.
An Annie Mayes aged 70 died Lisburn District 1868 but the image is pay to view on GRONI.

Whilst C of I registers for Blaris, Lisburn exist going back to the 1600's I don't think there are any 1800-1860 Presbyterian ones if check PRONI's Guide to Church Records. It is difficult to guess quite where he might have been baptised.
When there is an absence of local records there is a tendency on online trees to consider (or just link) to names and ancestors further afield.
John Mays senior may have married c.1825 in the C of I, as Presbyterian marriages were only legally recognised from 1845 onwards, but if it were located it won't list his father's name. Due to the Penal-laws his baptism will be undocumented if Presbyterian. Breaking into the 1700's is unfortunately only accomplished by a minority.

https://www.genuki.org.uk/big/irl/ANT/Derryaghy
http://lisburn.com/churches.html

John and Isabella at 9 Drumsill, Magheragall / 28 Trummery 1911

the 9 and 28 are just the reference code the enumerator used on his various forms - householders A, building B1 & B2 & summary N. They are not a house number or part of their 'address' and can be disregarded. The house number where there is one in towns appears on the 1911 additional page - the reverse of the householders form.

Thanks Rosie, all it needs is sunlight, photographed a different time of day or a mirror to reflect that onto it, I live 5½ to 6 hours away.

For the 6 counties the 1937 census is the first since 1911 and is held by PRONI.
https://www.bbc.co.uk/news/uk-northern-ireland-22848416
Would be handled separately like Scotland and likely require legislation by the NI Assembly following any changes passed in England & Wales.
It was taken under the Census (Northern Ireland) Act, 1936 and current ones are taken under a 1969 local Act https://en.wikipedia.org/wiki/Census_Act_(Northern_Ireland)_1969

http://www.histpop.org/ohpr/servlet//View?path=Browse/Legislation%20%28by%20geography%29/Ireland&active=yes&mno=4042

A census had been taken 1926 but householder returns do not survive (BBC link). It was intended to take one in 1941, so the 1937 omitted some questions (eg those regarding occupation and industry).
https://www.nisra.gov.uk/statistics/2001-and-earlier-censuses/1937-census

No proof was required a verbal statement. Proceedures date back to 1864 (based on 1837 in England) when many were illiterate and people didn't retain certs. Apart from Registry Office marriages the Registrar did not have access to the most recent marriage details to check facts as the books were in use in the churches and only deposited with him when full.

A person can change their name if they desire, eg use a step-fathers or adopted father.
https://www.gov.uk/change-name-deed-poll

There is no White-Hyslop marriage in England or Scotland either.
For an unmarried couple (ie Mary Jane says Mr White is the father but she is not married to him) the Registrar would leave the boxes blank, unless the father was present alongside her and jointly signed the birth entry.

Mr White could have been local or eg USA army stationed in Belfast who promised to marry and reneged. For a brief period she used the White surname but he never returned and James never having known his father disliked having the surname on his birth cert so reverted to his mother's. But the White formerly Hyslop might also be explained by separation of her mother Ellen from her father James and being raised by a Mr White (divorce was rare 1920-40).

Perhaps find the deaths/burials/death notices of Mary Jane's parents https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1920/01202/1500328.pdf to elimate some things.
You have the address on the birth can also look at the street directories and see who was there https://www.lennonwylie.co.uk/ but the address was her parents house and not hers.

As James & Ellen Hyslop were R.C. https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1918/09725/5528903.pdf and do not appear on the Belfast Council website they are likely interred in Milltown Cemetery with their death notices in the Irish News also not online, but deaths are viewable for £2.50 on GRONI to 1974.