Is it an appositional relationship (An t-Athair Ó Tuamtha) or an ownership one?

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__̴ı̴̴̡̡̡ ̡͌l̡̡̡ ̡͌l̡*̡̡ ̴̡ı̴̴̡ ̡̡͡|̲̲̲͡͡͡ ̲▫̲͡ ̲̲̲͡͡π̲̲͡͡ ̲̲͡▫̲̲͡͡ ̲|̡̡̡ ̡ ̴̡ı̴̡̡ ̡͌l̡̡̡̡.___

Both
It's an appositional genitive: Sráid Uí Chonaill

On a side note, sometimes the nominative is used instead of the genitive as a mark f respect.

This is particularly prevalent in religious contexts, e.g. the naming of churches etc...

There was a debate a good number of years ago to whether or not the genitive should be used with regard to Peadar - i.e. saothar Pheadar Uí Laoghaire or saothar P(h)eadar Ua Laoghaire - such was people's respect for the man.

Cian

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Please wait for corrections/ more input from other forum members before acting on advice

I see.

What would happen if you were talking about a sept, in a historical context?

Bealach Uí(bh) Néill -the avenue of the Ó Néill sept

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__̴ı̴̴̡̡̡ ̡͌l̡̡̡ ̡͌l̡*̡̡ ̴̡ı̴̴̡ ̡̡͡|̲̲̲͡͡͡ ̲▫̲͡ ̲̲̲͡͡π̲̲͡͡ ̲̲͡▫̲̲͡͡ ̲|̡̡̡ ̡ ̴̡ı̴̡̡ ̡͌l̡̡̡̡.___

rather:
Bealach Ó Néill
Bealach Ó gConaill
7rl.

Yes, you'd see this in the dative as well:
Mar gheall ar An tAthair Peadar Ua Laoghaire

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

Why the urú? Genitive plural?

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__̴ı̴̴̡̡̡ ̡͌l̡̡̡ ̡͌l̡*̡̡ ̴̡ı̴̴̡ ̡̡͡|̲̲̲͡͡͡ ̲▫̲͡ ̲̲̲͡͡π̲̲͡͡ ̲̲͡▫̲̲͡͡ ̲|̡̡̡ ̡ ̴̡ı̴̡̡ ̡͌l̡̡̡̡.___

Yes and Ó is the genitive plural of Ó, dative plural Uíbh.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

Both sounds strange.

I'd expect:
An tAthair Peadar Ua Laoghaire
saothar an Athar Peadar Ua Laoghaire
mar gheall ar an Athair Peadar Ua Laoghaire

Those are indeed correct, but native speakers, at least in Munster, do retain the nominative out of respect. It also often happens for pen names, but that's probably not as surprising.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal