Potential repercussions from overlapping university appointments?
From the standpoint of the law it should be fine. Most (close to all?) countries allow you to hold several appointments in the same month. If not, some part-time workers (think waiters, bartenders, etc.) would have a hard time making ends meet.
The other legal factor is the employment contracts. These could require you to seek authorization from the current employer before starting outside employment.
It absolutely is a conflict of interest, but one that should be possible to solve by disclosing the situation to both employers. The actual solution might be the one-month leave you mention, quitting the first position earlier, getting a thumbs up from the current employer, or doing a 50/50 time split the first month. Either way, the current institution deserves to know if you begin to work for, and earn money from, a new institution - whether you intend to leave your current job or not.
Note that the possible conflicts of interest don't just concern the pay - if you use resources from the first institution to benefit the second it can also be bad. This isn't really solved by a leave, but by disclosure and then handling the resources responsibly.
I don't really see any professional issues, assuming you handle the conflict of interest issues professionally (i.e. disclose them). After all, it is rather common in academia to have more than one affiliation. Of course, one also needs to be professional in choosing which affiliation to put on papers.
Another consideration, especially in the US, would be benefits and medical insurance. The easiest way to handle this is probably to waive those benefits for a month at the new job. And if the two institutions are in different jurisdictions you might have a fun time figuring out how to do your taxes correctly...


Another benefit of disclosing the situation properly is that you can then ask HR (at both institutions!) if there are other relevant issues in the specific case. Ultimately, it's their job to know how to handle these things, not yours.