A convict ship sailed from Cork in November 1802 that arrived in Port Jackson in March 1803. Now the song "Back Home In Derry" starts:

In 1803 we sailed out to sea
Out from the sweet town of Derry

Many thanks for confirming what was stated earlier, i.e. that no convict ship sailed "out" from anywhere in the British Isles in 1803, especially none from "the sweet town of Derry".

"Back Home In Derry" is total fiction and has no basis in any historical event.

To answer your question - what was the significant event in 1803 in the northern province near Derry that 'back home in Derry' is really all about?

The short answer is that there were none. A bit further afield the only thing I can turn up is Robert Emmet's Non-Rebellion of 1803 - otherwise named by the man himself as "the dog that didn't bark."

Robert Emmet’s verdict on the Dublin insurrection—‘there was failure in all: plan, preparation and men’. This applied to an even greater extent to events in Ulster. The attempt to raise the North produced no battles, or even skirmishes; in fact not a single shot was fired in anger. The areas of Ulster the United Irishmen tried to raise were in the counties of Cavan, Fermamagh, Down and Antrim. Derry doesn't seem to have featured at all.

The First Fleet arrived in Botany Bay in 1788 but the location was considered totally unsuitable. Botany Bay was rejected as a place of settlement and Port Jackson (Sydney) was chosen instead. It is still considered to be the greatest natural harbour in the world. In 1803 the convict colony in Tasmania was established, followed by one in Queensland in 1824. Western Australia was established as a "free colony" (i.e. it did not receive any convicts) in 1829 but in 1850 due an acute labour shortage it began to receive convicts around the time that the states on the east coast of Australia stopped taking convicts. Transportation as a means of dealing with felons peaked in 1830 after which it went into a steep decline finally ending in 1868.