I am a UK citizen, and I was a J1 visa holder as a postdoc for 2.5 years in the USA. I claimed the UK-US tax treaty and paid no tax in the US. However, because I overstayed my 2 years, and therefore had to pay the back tax back. Now I am a professor in the UK, but I am visiting the US again, on another J1 visa for 2 months this summer as an exchange scholar. Can I claim the UK-US tax treaty again? If so, does this mean I should declare my US earnings to the UK tax authorities (or does it mean my US earnings are completely tax free in both the US and UK). Any info appreciated. Thanks.

The exemption on J-1 is for two years. If you exceed the two years period, the exemption is lost for the entire visit including the two years.
The exemption is available only once. So another visit for 2 months is not exempt from federal and state income taxes as well as FICA taxes.

QUOTE,"However, because I overstayed my 2 years, and therefore had to pay the back tax back. "--- Right!Just for reference, the provision is called"the retroactive tax clause" that is contained in many of the tax treaties. The provison states that, if the J-1 visa holder EXCEEDS the specified tax exemption time limit(in this case for two yeas as an exchanging scholar), then the tax exemption is void and all federal income taxes including the interest due on the late tax payments become due retroactive to the date of arrival.
So, as a J1 visa holder, overstaying the regular 2 year period means paying back taxes as well as the fines retroactively for your income that you earned over the two regular years(though this clause CAN be waived) but only on the income that is U.S.-sourced including state income taxes( if applicable).