The EU also has the Charter of Fundamental Rights of the European Union. The final court for that is the European Court of Justice (or Court of Justice of the European Union) in Luxembourg. The ECJ is a EU organ, and appeals instance for violations of EU law in general. As such, since the ECHR is party of EU law by treaty, the ECJ also handles cases that involves the ECHR.
Once the UK leaves the EU, it will no longer be bound by judgements of the ECJ, but it will be bound by judgements of the ECHR.
The UK Human Rights Act sets out the obligations for the government with respect to the ECHR. The ECJ obligations I believe are indirect via the European Communities Act 1972, which gives EU law and treaties primacy over UK law.
Indirectly, leaving the EU does make it legally possible for the UK to withdraw from the ECHR, but that would still mean leaving the Council of Europe as well, but that's unlikely - it would put it in company with Kazakhstan, Belarus and the Vatican City as the only European states which are not CoE members (the Vatican is an observer).