The veteran peace campaigner Brian Haw faces eviction from an area of grass in Parliament Square Gardens after losing an attempt to launch a legal challenge against a possession order granted to the mayor of London.

Haw's longstanding presence on the pavement on the east side of Parliament Square is not, however, threatened by the order, which relates to his encroachment on to a small adjoining part of the gardens where he has pitched a tent. Haw has come under pressure to quit his decade-old protest just metres from Westminster Abbey as the royal wedding approaches.

After the appeal attempt was lost, co-campaigner Barbara Tucker said she did not believe the eviction had anything to do with clearing the area for the royal wedding – "it is about getting rid of our peace campaign".

Last month, the mayor, Boris Johnson, won a high court possession order to evict Haw and Tucker.

Rejecting the application at the court of appeal, the master of the rolls, Lord Neuberger, and Lady Justice Smith ruled there was "no prospect" of any appeal being successful. They said the mayor was entitled to his order for possession without any further delay as "justice delayed is justice denied".

Tucker interrupted Neuberger as he gave his judgment, saying: "This is a cover-up. Are you finished now?"

She and several supporters walked out of court as Smith announced that she agreed the application should be dismissed. Haw is being treated for lung cancer in Germany.

Mr Justice Wyn Williams last month granted an order for possession and an injunction against Haw, from Redditch, Worcestershire, and Tucker, but the orders were not to be put in place until after any appeal.

All Haw and Tucker have to do now, however, having lost the right to appeal, is to move their tents from the green, owned by the Greater London Authority, to the pavement, which is owned by Westminster council.

The prime minister, the home secretary and the mayor of London have all vowed to clear the pavement of protesters before Kate Middleton and Prince William marry on 29 April, but so far have found no legal power that allows them to do so.

The high court judge ruled: "Parliament Square Gardens [PSG] is not a suitable location for prolonged camping; such camping is incompatible with the function, lawful use and character of PSG and it is also inconsistent with the proper management of the area as a whole."

He said the campaigners' tents and placards were occupying more space than was permitted. The pair would be allowed use of a three-metre length of kerb to display placards as that had been a part of the protest for several years, the judge said.

Last July, bailiffs and police evicted demonstrators from Democracy Village – the scattering of tents, placards and home-made police boxes set up in the square in May 2010 – after the mayor was granted a possession order for the site, citing vandalism. However, the court remitted the question of whether it was reasonable and proportionate to enforce orders against Haw, whose decade-long presence on the pavement on the east side of Parliament Square was not challenged, except for his encroachment on to a small part of the gardens.

Westminster council has launched legal action against protesters who occupied the footpath after being evicted from the green last year.