Excuse me for warming this thread up so late, but I've just joined the forum, and found your posting very interesting.
First of all, your "Wildwood Flower" sounds just fine. But then, it's not really a folk song. It may be of American origin, but it's an art song, and the lyrics are written in standard English, so any British, American or Colonial accent would fit. The song only got its aura of folksiness because Maybelle Carter, whose recording made it popular, had a very strong regional accent. So unless you want to cover Maybelle's version, you don't have to copy her accent. (And if you did want to cover it, you'd also have to copy her guitar picking, which is at least as definitive as her singing!)
Songs with lines like "I ain't got no-one to love," on the other hand, sound silly when sung with an Irish, Scots or English (except Cockney) accent, because "ain't" and the double negative just aren't part of those languages, and they jar.
So my advice would be to be guided by the lyrics of the song itself, rather than by the accent of the recording artist you've learnt it from.
I'm an Ulsterman born and bred, so the bulk of my songs are Irish, which is no problem. And I spent 6 years of my childhood in Scotland, so I've got the clear vowels and rolling "r" that Scots songs call for.
My take on songs from other parts of the English-speaking world is to try them in Received Pronunciation (with my slight Ulster-Scots colouring) and see whether I trip over some dialect word that doesn't sound right in RP, or some rhyme that doesn't work. If I do, I try to sing a "light" version of the dialect in question. Not everyone can do this - it's a gift that runs in my family. My father could mimic accents, so can I, and so can my son. My son and I have the added advantage of having been immersed in different dialects of our mother tongues (he in German, I in English) during childhood. But it's something you can work on, in addition to your vocal and instrumental skills.
I have few problems with songs from your country, like "Waltzing Matilda" or "The Dying Stockman". They contain "Australianisms" like "billbong" and "dingoes", but these are not dialect words as such - they're the Oz words for Oz phenomena for which British English has no equivalent, so they've found their way into standard English, and they don't jar on the ear. (Well, they might jar on an Aussie ear! When I sing these songs, I ask any Aussies in the audience to imagine that I'm an Irish immigrant who has turned swagman or stockman.
Songs by Robert Burns can be a problem when sung by non-Scots. Some of them can be "anglicised" (like changing "An' I sae weary, fu' o' care" to "And I so weary, full of care"), but many contain so much Broad Scots vocabulary that this is not posible. And using the accent of one language with the vocabulary of another really is incongruous.
There are lots of songs that refer to definite places in specific regions or countries (Scarborough Fair, Star of the Co. Down, Loch Lomond ...), but I approach these like art songs - if they're written in standard English, the accent doesn't matter. After all, you could be a holiday-maker who has fallen in love with the place mentioned.
Have fun with your international repertoire. Make each song your own, and then you won't have to worry about the accent!