At present, we just don't know for sure.
Time measurements are based on Earth standards as far as we can apply them to time in Star Wars.
There is a specific event in the season finale which is of importance within the Star Wars universe, although likely unknown to the galaxy at large at the time;
! The destruction of Tipoca city by the Empire.
This may have a date canonised at some point in the future, which would allow us to pin point how much time has passed since Order 66 was issued, but at present that's not happened as the Bad Batch finale was the first depiction or mention of that event.
The show itself doesn't allow us to clearly track time. Although the events of each episode is not shown to last more than a day or maybe two, there are cuts where the squad are shown travelling between system. The duration for interstellar travel times in Star Wars are poorly defined, often described as mere hours to cross a large part of the galaxy.
On top of that there are events which happen off-screen, usually mentioned in the opening scene of each episode. We don't really get to know how long these events take, but given the context I wouldn't expect those breaks to be more than a week or two. 963
Contrary to this, there is plenty of development happening within the Empire, with the introduction of TK troopers. The Empire might be rushing this as much as possible, given the distaste held by <PERSON> in particular against clones, but it's not something they could implement overnight, at least not to the degree shown by the end of the season.
Finally, we do have a character who might offer some guidance. <PERSON> is still a child and isn't shown to age visibly through the course of the season. While this may be down to artistic reasons (for comparison, <PERSON> aged through occasional redesigns of her character in Clone Wars), we could take this as evidence that the events of the first season couldn't have lasted longer than a year.
All this taken together, unless and until dates are given in future works, we could expect the season finale to have occurred between two and twelve months from Order 66. My personal opinion, based on the story progression and development, would be towards a shorter time period of between two and four months. 503