It’s an open question, the extent to which SARS-CoV-2 (that’s the virus that causes COVID-19) infects the brain as well. There is evidence from SARS, that’s the epidemic from 10 years or so, or a bit more than that, ago that that does infect the brain a bit and may actually affect the respiratory centre. That’s the part of the brain that drives your breathing and may be one reason why people have their breathing so badly affected.
There is no evidence that when you’ve had COVID-19 there’s any lasting effects on the nervous system or indeed that it does infect the nervous system. But, if it’s like SARS it may well do but it will be a temporary phenomenon.