Birth Records

Those dates seem a bit odd, since Phillip would have been over 60 when he married.  In any case, civil birth records didn't start until 1864, so there won't be any to find for that couple.  Baptismal records may be available, though.

You didn't mention whether your Campbell's were Protestant or Catholic.  Many Presbyterian Campbell's came to Ireland during the Plantation of Ulster, though mostly to Ulster itself.  However, there were Campbell's who migrated to Ireland before the Reformation, whose descendants remained Catholic since they had intermarried with the "native" Irish.  Some of them migrated to Mayo and Galway (including some of my ancestors from a few parishes to the north of Tuam, in Mayo).  In case yours were Catholic, the Catholic parish registers for Tuam which are available online go back to 1790 (with a few gaps), and can be accessed at this link:

https://registers.nli.ie/parishes/1082

Unfortunately, one of the gaps is 1804-1810, but you may be lucky and find your Phillip Campbell nonetheless (and, as I said, that seems a suspicious birth year anyway).  In those days, most people had only a vague idea when they were actually born (and no written record of it), so later records of ages or birth years need to be treated with caution.

Nelis & Nellis are common anglicizations of the Irish surname Mac Niallghuis, which originated in Donegal.  Many people migrated to Mayo and Galway form Ulster during and after the Plantation of Ulster, though (again, including some of my realtives), so it wouldn't be surprising to find some of them in the area of Tuam.

Note also that, if what you have is a civil marriage record, the place name(s) on it can be confusing.  Civil records were organized by registration districts, which were usually named for the largest town or city in the area and could cover a number of parishes.  The certificate may also give a townland and/or parish name, and that would be much more useful in figuring out where the family actually lived and finding further records.

Going by the marriage information you gave...........this may be the family.    Birth year of 1810 error or another Philip Campbell.
Look like this one in Donegal.

Please Check.

The Marriage
8126873.pdf (irishgenealogy.ie)

1901 Census 
census.nationalarchives.ie/reels/nai000632243/

Philips death
4545184.pdf (irishgenealogy.ie)

1911 Census

National Archives: Census of Ireland 1911