I'm not clear on what the title of the thread is based. The two regiments, the 43rd and the 44th, were entirely separate and shared no common lineage. Do you think that William Cathcart served in both Regiments, or are there two separate individuals named William Cathcart, one in each regiment, either of whom might be the man you are interested in?
Furthermore we don't know what period of service we should be looking at. If WC was born 1765 he might have been expected to join the Army circa 1783 and then, depending on his health etc serve for up to 20 years. That means that when the son was born in 1803, WC would either be just about to end his service or already have left.
Turning to the Irish connection, the 43rd (Monmouthshire) Regiment of Foot (Light Infantry) to give them their full title, were only in Ireland in 1787-93. Although the Regiment had a second battalion this wasn't raised until around 1804 and then only served in Europe as part of the Napoleonic Campaign.
The 44th (East Essex) Regiment of Foot were stationed in Ireland in 1792-3 and then went to the West Indies and didn't return to Ireland until 1802. That Regiment's second battalion was raised in Ireland in July 1803 and then moved to England the following year. Don't be misled into thinking that just because WC was born in Essex, he would have automatically joined his county regiment. As can seen by the fact the 2nd battalion was raised in Ireland, Regiments got recruits wherever they could find them.
And then lastly, the Tyrone Militia was an entirely separate unit, not affiliated to either of the two regular army regiments mentioned so far. Their role was the internal defence of Ireland and most militiamen only served for about 3 years, and even then they were part timers. With the rise of the threat from France at the end of the eighteenth century some militiamen were encouraged to join the regular army and so might have, for instance, joined the 43rd or 44th if those regiments were recruiting in Ireland at the time. However the military threat to Britain from Napoleon didn't really start until 1793, by which time WC would have been aged 28 - comparatively old to join the Army, but a more normal age to join the Militia which tended to favour single men or married men with no children under 10 years of age.
