(1) Background: High blood pressure (HBP), overweight, and obesity are common, growing public health problems worldwide. The aim of this study was to evaluate associations between changes in body weight status and HBP among Lithuanian children and adolescents during the COVID-19 pandemic. (2) Methods: In this study, we analysed data on blood pressure and anthropometric measurements of 2430 children and adolescents aged 8–18 years, who participated in both the baseline study conducted before the COVID-19 pandemic (from November 2019 to March 2020) and the follow-up study during the COVID-19 pandemic (from November 2021 to April 2022). Multivariate logistic regression analysis was used to estimate the associations between changes in weight status categories and HBP. (3) Results: At baseline, 17.1% of the subjects had overweight, 5.9% had obesity, 5.6% had abdominal obesity, and 23.7% had HBP, whereas at the follow-up, these percentages increased to 20.1%, 8.2%, 6.8%, and 27.4%, respectively. Compared to schoolchildren who maintained normal weight from baseline to the follow-up period, subjects who newly developed overweight/obesity and those who remained with persistent overweight/obesity had increased odds of HBP, with adjusted odds ratios (aORs) of 1.95 (p < 0.001) and 2.58 (p < 0.001), respectively. In subjects who transitioned from overweight/obesity to normal weight, the odds of HBP were slightly increased, with an aOR of 1.14 (p = 0.598), but the change was not statistically significant (p > 0.05). (4) Conclusions: This study observed an increase in the prevalence of overweight, obesity, and HBP among schoolchildren during the COVID-19 pandemic. The study also suggested that changes from normal body weight status at baseline to overweight/obesity during follow-up, especially persistent overweight/obesity, were associated with higher odds of HBP in Lithuanian children and adolescents during the COVID-19 pandemic.