In 2016, United Kingdom (UK) citizens voted to leave the European Union (EU), which was officially implemented in 2020. During this period, UK residents experienced a great deal of uncertainty around the UK's continued relationship with the EU. Many people have used social media platforms to express their emotions about this critical event. Sentiment analysis has been recently considered as an important tool for detecting mental well-being in Twitter contents. However, detecting the psychological distress status in political-related tweets is a challenging task due to the lack of explicit sentences describing the depressive or anxiety status. To address this problem, this paper leverages a transfer learning approach for sentiment analysis to measure the non-clinical psychological distress status in Brexit tweets. The framework transfers the knowledge learnt from self-reported psychological distress tweets (source domain) to detect the distress status in Brexit tweets (target domain). The framework applies a domain adaptation technique to decrease the impact of negative transfer between source and target domains. The paper also introduces a Brexit distress index that can be used to detect levels of psychological distress of individuals in Brexit tweets. We design an experiment that includes data from both domains. The proposed model is able to detect the non-clinical psychological distress status in Brexit tweets with an accuracy of 66% and 62% on the source and target domains, respectively.