To date, multiple efforts have been made to use genome-wide association studies (GWAS) to untangle the genetic basis for SARS-CoV-2 infection susceptibility and severe COVID-19. However, data on the genetic-related effects of SARS-CoV-2 infection on the presence of accompanying and long-term post-COVID-19 neurological symptoms in younger individuals remain absent. We aimed to examine the possible association between SNPs found in a GWAS of COVID-19 outcomes and three phenotypes: SARS-CoV-2 infection, neurological complications during disease progression, and long-term neurological complications in young adults with a mild-to-moderate disease course. University students (N = 336, age 18–25 years, European ancestry) with or without COVID-19 and neurological symptoms in anamnesis comprised the study sample. Logistic regression was performed with COVID-19-related phenotypes as outcomes, and the top 25 SNPs from GWAS meta-analyses and an MR study linking COVID-19 and cognitive deficits were found. We replicated previously reported associations of the FURIN and SLC6A20 gene variants (OR = 2.36, 95% CI 1.31–4.24) and OR = 1.94, 95% CI 1.08–3.49, respectively) and remaining neurological complications (OR = 2.12, 95% CI 1.10–4.35 for SLC6A20), while NR1H2 (OR = 2.99, 95% CI 1.39–6.69) and TMPRSS2 (OR = 2.03, 95% CI 1.19–3.50) SNPs were associated with neurological symptoms accompanying COVID-19. Our findings indicate that genetic variants related to a severe COVID-19 course in adults may contribute to the occurrence of neurological repercussions in individuals at a young age.