My question is this: how would people detect the 1 year time jump?
if you send it directly between the star systems not using the wormhole (sol to A-Centauri or vise-versa), it will take the 4.5 or so lightyears
This means that if I sit on the Sol side of the wormhole, and I send a signal through it, I would hear that signal again after 5.5 years (1 year due to wormhole, 4.5 years for normal travel from AC to Sol).
That will raise some eyebrows, as scientists will know AC is 4.5ly away.
When you come back, you are 1 year in the past (relative to A-Cent).
Furthermore, if you repeat the same experiment while sitting on the AC side of the wormhole, it will take 3.5 years to hear the signal again (-1 year due to wormhole, 4.5 years for normal travel from Sol to AC).
While the first test may have been shrugged off as some unexplained delay (e.g. the signal passed through some obstacle like a gas cloud at a speed lower than the speed of light), this second test would raise major eyebrows, as it seems to suggest that the signal traveled faster than the speed of light (4.5ly in 3.5 years), which is a big thing to discover and will not be shrugged off.
After that, further refined testing could be done to explain how this signal managed to reach back in only 3.5 years of time. There are several ways to do this: star positioning, elemental decay, some sort of permanent radio transmission which allows you to figure out what time it is (relative to the source of the transmission), celestial events that are observable from both Sol and AC and whose distance to both is known precisely, ...
These refined tests are very niche and not something you would naturally stumble upon, but the initial observation I mentioned is something that you could serendipitously stumble upon (e.g. if a certain event caused a very unique signal, and some random protagonist just happens to record the loopback after 3.5 years and then realizes it's only been 3.5 years since "the event", thus triggering them to contact others about their findings)