EU law takes effect within the UK by operation of the European Communities Act 1972 and any other specific implementing legislation: see s. 18 of the European Union Act 2011 and Pham  UKSC 15 at §80
The UK retains sovereignty both through Parliament but also as a matter of democratic oversight through elections and referenda and in its prerogative over fundamental issues such as foreign and security policy.
The scope of EU law remains limited by the terms of the EU Treaties and the principles of conferral and subsidiarity.
Outside that scope, UK law is unaffected by EU law – for example, UK education, health and criminal policy are very largely unaffected by EU law.
Within the scope of EU law, the principles of supremacy and direct applicability of EU law were well established before the UK joined the European Communities.
Although each of the Member States, including the UK, can be outvoted on certain measures by a ‘qualified majority’ of the other Member States, this is relatively rare and the UK has the right to challenge the legality of such measures before the ECJ.