Another issue is the seriousness of the song. I tend to use accents with funny songs but not otherwise.
So I sing 'Robert Emmet' & 'Roddy McCorly' naturally But I can't seem to sing 'Begorrah' or 'Old Orange Flute' without a fake accent.

Same applies to London songs. I will sing 'Streets of London' naturally but will lapse into my native cockney (& perhaps even exaggerate it a litte) for 'British Workman's Grave'

Bert.