If Theresa May requests a three month extension of Article 50 (or longer), then the European elections will take place while the UK is still an EU member.
There are many reasons why people voted to leave in 2016, but they did not include allowing the UK government to abolish democracy.
British citizens and EU27 citizens in the UK (who choose to vote in the UK) must have representation in the European Parliament through their MEPs.
Our representatives are elected and not appointed.
The Financial Times has reported that if the Article 50 divorce process were extended, and the UK was still negotiating its exit from the EU in May, Britain would have no choice but to participate in the European polls.
UK faces prospect of taking part in European elections, Financial Times, 14 March 2019
The Financial Times article quotes the Electroal Commission as saying that legislation requiring these elections has not yet been repealed.
Cmmenitng on the FT article, Roger Casale, FOunder and CEO of New Europeans said:
"This is techically correct, but in reality rather misleadng because as things stand the legislation will be repealed automatically on 29 March even if the UK is still a member of the EU unless the Government acts."
Schedule 9 of the EU (Withdrawal) Act 2018 lists the European Parliamentary Elections Act 2002 and the European Parliament (Representation) Act 2003 among the Acts to be wholly repealed.
Unless the Government changes the specification of the exit day in Schdedule 9, curently given as 29 March, the legilsation will be repealed on that day come what may.
Not taking part in the European elections could mean that the UK would have to leave the EU with or without a deal if Brexit is delayed beyond 1 July.
The new European Parliament will convene from 2 July and it is a requirement that all EU citizens are represented.