This is potentially big enough to influence the climate for a year or two. We need more information (amount of SO2 released, tephra / DRE volume, and confirmation on how high the plume went to confirm stratospheric injection of those particulates), but it's looking like this has the potential to be a VEI-6 around the size of Pinatubo based on the discussions of several volcanologists/geologists that I follow, which caused a cold climate anomaly for 1992 (and also decreased SST's a bit). Edit: Looks like it was rated VEI-5, so right around just big enough to influence the climate but probably not to the extent of Pinatubo unless it was very high in SO2.
I'm not an expert but I believe volcanic climate forcings are usually restricted to the hemisphere of the eruption outside of the really truly massive ones, so I'm not sure if we'll see a global anomaly or centered on the southern hemisphere.
This is fascinating however and completely came out of nowhere. Very, very large and extremely powerful eruption with little to suggest a caldera forming event was in the pipeline before it happened. The smaller phreatomagmatic / phreatoplinian eruptions were impressive, but this was just another level with the pressure wave circling the globe, the loud explosive crack being heard at great distances a la Krakatau, and the tsunami generation.
Edit: Here's an incredible summary
https://www.severe-weather.eu/news/tonga-volcano-massive-eruption-explosion-stratosphere-usa-tsunami-shockwave-fa/