Background: Some studies have identified declines in mental health over the course of the COVID-19 pandemic across the world and in different age groups, including older people. As anxiety and depression are common neuropsychiatric symptoms among people with dementia or mild cognitive impairment, the mental health experiences of older people during the pandemic should therefore take cognitive function into consideration. This should also be examined using quantitative measures that were assessed prior to the pandemic. This study addresses such gaps in the evidence base on depression and anxiety among older people with cognitive impairment before and throughout the COVID-19 pandemic. Methods and Findings: Using data from the English Longitudinal Study of Ageing (ELSA) collected from 2018/19 to Nov/Dec 2020, we estimated changes in depression and anxiety for people aged 50+ in England across three cognitive function groups: no impairment, mild cognitive impairment, and dementia. We found that depression (measured with CES-D score) worsened from 2018/19 to Nov/Dec 2020 for people with mild cognitive impairment (1.39 (95%CI: 1.29-1.49) to 2.16 (2.02-2.30)) or no impairment (1.17 (95%CI: 1.12-1.22) to 2.03 (1.96-2.10)). Anxiety, using a single-item rating of 0-10 also worsened among those with mild cognitive impairment (2.48 (2.30-2.66) to 3.14 (2.95-3.33)) or no impairment (2.20 (2.11-2.28) to 2.85 (2.77-2.95)). No statistically significant increases were found for those with dementia. Using a clinical cutoff for likely depression (CES-D>=4), we found statistically significant increases in the probability of likely clinical depression between 2018/19 and Nov/Dec 2020 for those with no impairment (0.110 (0.099-0.120) to 0.206 (0.191-0.222)) and those with mild impairment (0.139 (0.120-0.159) to 0.234 (0.204-0.263)). We also found that differences according to cognitive function that existed before the pandemic were no longer present by June/July 2020, and there were no statistically significant differences in depression or anxiety among cognitive groups in Nov/Dec 2020. Conclusions: Our findings on measures collected before and during the pandemic suggest a convergence in mental health across cognitive function groups during the pandemic. This suggests mental health services will need to meet an increased demand that will come from older adults, especially those not living with cognitive impairment or dementia. We also found little significant change in mental health outcomes among those with dementia; as their existing need for support will remain, policymakers and care practitioners will need to ensure this group continues to have equitable access to support for their mental health.