Are you liable to pay taxes if you no longer hold a resident alien card but receiving a social security pension plus a private pension which combined is still under the tax allowance. Thank you

If BOTH the Social Security and private pension distribution (reported on Form 1099-R) is still under the threshold for taxable income, then you have no legal requirement to file a U.S. tax return, which, for a U.K. citizen, would be Form 1040NR.