I actaully favor separate timelines, the origininal movie being the stand alone, however, if you theorize that there was never just going to be one gathering, that there have, if fact, been many over the centuries - perhaps one every couple thousand years or so, then you might begin to be able to build a single time line. Not every immortal attends gatherings, and resists the call, new immortals are always being created even during a gathering as seen in the series, therefore, a future gathering would be the more logical endgame for a noob  than the one depicted as occuring in the first season of the series.  A cyclical timeline would go a long way to resoving your issues, don't you think?

The creators of the film assumed a closed story in 1-3. When the idea of the series came into being, one can assume that the Gathering known from films was the destiny of Connor himself. Jacob Kell had a similar target - kill Connor.

Concept aside, in pure mechanics is there a conflict? Like are we told word-of-god in some on-screen epilogue that Conor was the last, or did characters in the film merely express that opinion?

In the original film Connor was the winner of the prize. Period. Connor at the very least indicates as much upon beating the  Kurgan. For the first movie,  there was no Duncan, no alternate plot lines, so the original movie functions on the assumption that Connor really was the last immortal on Earth after he killed the Kurgan and really did win the prize. The lure of franchise money, however, led later to weird convolutions including fan theories that the winning of the prize shattered reality as known and spun off multiple realities such as the tv series timeline.