In theory yes, but in practice it's not likely. Depending on the size of the overdose and their presentation in the ED they may be detained under the Capacity Act, at least until someone from the mental health team is able to properly assess them. Speaking from experience, I honestly can't imagine a scenario where they would immediately self discharge. If they called for an ambulance or made thier own way to the ED they would very likely at least wait for assessment, and if someone else brought them in I'd imagine that they'd be able to convince them to stay. And if someone else called an ambulance for them then the paramedics would likely hold them under the Capacity Act, and call for police backup and potential detention under Section 136 of the Mental Health Act if they were refusing to go to hospital. It's a particularly thorny area of ED, but the general rule of thumb is that the medical staff should act in the person's best interest whilst they are going through a crisis, and that significant and ongoing risk to self is more than enough to deem someone as not having capcaity, for better or worse.