It depends on what aspect you are looking for (the question is a bit vague ...):

The Earth always had jet streams, for example, or trade winds, although their characteristics change with orbital parameters. Other more detailed properties (typically smaller scale) would depend on local topography, which would have varied over the geological past. Examples here would be monsoon circulations or more detailed structure in eddy driven jets. The shape of ocean basins has an influence on the ocean heat transports, for example, which in turn would modify the overlying atmosphere. The size of ice sheets (think glacial cycles) has big effects on local winds but more profoundly on the hydrological cycle. So there are many details to consider, some of which are always present but nearly all of them would be modified by the precise boundary conditions.

References to research papers would only address specific issues. Any book on atmospheric dynamics would be helpful, but possibly Ray Pierrehumberts book on planetary atmospheres might give a more global view. I am not particularly an expert on paleo-climate literature, but I would imagine there are textbooks out there to start with.

As soon as there was an atmosphere on earth! I don't think it is a question of  composition of the atmosphere. But rather of  gaz concentration. A "wind" can exist even with low to very low concentration of gaz. Thus wind circulation (local to global scale) exists since the atmosphere could be detected on earth.

I do agree with KM Towe on the rather large initial question. I think that the right question would have been: at which geological time have we concordant proofs that any orgenized atmospheric circulation existed on earth? A question quite difficult to answer... Generally past weather reconstructions (let us say of large atmospheric systems eg El Nino, Monsoons...) transport us some millions years ago (quaternary, Tertiary). Farther is difficult to say to my own knockledge...