Joanne,
Explanation of Brethren here:
https://en.wikipedia.org/wiki/Plymouth_Brethren
In Ireland they generally don’t like to be referred to as Plymouth Brethren. If they have to be designated anything, most prefer to be called simply Brethren. Without getting bogged down in the details, many people have been attracted to the denomination because it offered a stricter interpretation of the bible than in the Church of Ireland or Presbyterian doctrines.
You’ll see from the 1911 census that 4 of the Reynolds household were Brethren then.
Brethren don’t have a Minister or anyone qualified to conduct a religious marriage service, and so get married in a Register Office. Some Register Office services in Ireland may because of the couple being of mixed denomination, but being Brethren is another category too.
Walter’s birth certificate seems to tie him fairly firmly to Co Down. His father was born in Co Down too and the family appear to have live dint hat area steadily from the 1840s onwards and they were still there in 1901. I think Walter’s story of being born in Wigan is probably inaccurate. It is possible he went to England to work in the coal mines (many people from Ireland did that) before he emigrated but he doesn’t appear to have been born there.
There was a John Reynolds farming in Ballyknock in 1833. Possibly George’s father? Anyway it shows the fmaily were well settled in that area. Consequently it;s very unlikely that one of their children would be born in Wigan.
http://www.irishgenealogyhub.com/down/tithe-applotment-books/moira-parish.php
Possibly the Ellis Island passenger born in Wigan isn't your Walter. Many migrants went to the US via Canada (as it was cheaper) and entered across the land border. Did Walter naturalise in the US? If so, does his application record where and when he first arrived in the US? (They often do).