Debra,
If your grandparents were Episcopalian then that?s an indicator their ancestors were Church of Ireland here. So I would certainly give the Glenavy Church of Ireland parish records careful scrutiny. (The only flaw in my logic is that most United Irishmen were either RC or Presbyterian, because those denominations suffered discrimination that members of the Church of Ireland did not. However there were other injustices and some COI did support the United Irishmen).
Birth certificates weren?t introduced in Ireland till 1864 so there are no Government records that you are going to find that will show their dates of birth, marriage dates or deaths. For the 1700s, there?s not much more than church records unless you were major landowner or notorious in some way.
Don?t expect too much from the church records even if you find them. Though there will be a baptismal date, for the late 1700s you often don?t get the birth date, and so all you can really say is the person was probably born within the previous 2 or 3 months. Sometimes you get townlands (addresses) but often you don?t. It?s pretty limited. Marriages don?t have parents names etc. However with Burt being such a rare name, you may be able to pick up other members of the family in the registers and piece them together.
You are correct that very few records go back to the 1700s. (Glenavy is one of the few parishes that does). As Clare has commented, research in the 1700s is difficult. Sometimes next to impossible.
It?s probably worth checking the tithe applotment records for Glenavy (which date to around 1830) to see if there are any Burts listed. (Only people with land would be there, so it?s not as good as a census, but better than nothing. Those for Co Down are not on-line and so again a trip to PRONI is required to view them. That might throw up any Burts that stayed in Ireland.
I don?t know of anywhere near Belfast called Glenarg. There?s Glenarm which is about 30 miles from Belfast, in Co Antrim, and obviously the parish of Glenavy. That?s probably the 2 nearest.
The defeat of the United Irishmen was in 1798, so if Nathaniel had to flee in 1801, it took him 3 years to get around to it. As I said previously, most of the rank and file were pardoned. Unless he was higher up in the organization he wouldn?t have had to flee (though he might well still have chosen to leave). The United Irishmen were created because of injustices in respect of Catholics and Presbyterians (restrictions on what they could do), so even after the defeat they had political and economic motives for leaving, together obviously with opportunities for cheaper land etc in the US and other destinations.
Elwyn