This is how the song sounds like:-
The lyrics are are as follows:-
wajaba al-shukru 'alayna,
ma da'a lillahi da'
Ayyuha al-mab'uthu fina
ji'ta bi-al-amri al-muta'
Ji'ta sharrafta al-Madinah
marhaban ya khayra da'
In English, they mean:-
O the White Moon rose over us
From the Valley of Wada'
And we owe it to show gratefulness
Where the call is to Allah
O you who were raised amongst us
coming with a work to be obeyed
You have brought to this city nobleness
Welcome! best call to God's way
So yes, I believe that when you hear this song, you are actually hearing it just like it was sung 1400 years ago. In the course of time, there had been many variations to the song, some even completely changing the tune to a nicer one while preserving the lyrics as is done by some Indonesians/Malaysians here:-
And what's this Olivia Newton John also singing the Tala'al Badru?
Well anyway that's my conclusion. Please feel free to correct me.