Mr Pound said TfL first gave April as the date for completion, then June, and now Mr Mahfouz says after asking for progress reports they have been told it will not be ready until September - at the earliest.
Mr Mahfouz said: "The contractor has screwed up monumentally - they are now saying September but even that they are saying is the earliest.
"We have already been let down and it keeps on slipping backwards.
"People who have accessibility issues are desperate to be able to get into central London. This sort of serious delay is an insult to them. Residents are fuming.
"Why did they leave it until a couple of weeks ago to say there's no hope of achieving the June deadline? We have no confidence in them delivering by September at all.
"I am calling on TfL to arrange an urgent on-site inspection to check progress and be transparent about when our long-awaited and much-anticipated lift will be delivered."
Mr Pound added: "We've been waiting since 1951 and that is long enough."
Ealing's cabinet member for transport said a list of various reasons were causing the delay, including - he thought- an issue with contractors ordering the wrong parts.
David Waboso, London Underground’s director of capital programmes, said: “Our work to deliver step-free access at Greenford Tube station remains on course to be completed later this year.
"During their work to install the new lift our contractors recently identified water leakage in the lift chamber. The issue has been resolved and work at the station is moving ahead as planned.
"When the lift comes into service Greenford station will be step-free for the first time since it opened in 1904, making journeys easier for wheelchair users and for anyone else who has difficulty using escalators and stairs.”
Mr Mahfouz said plans for a lift at Greenford were shelved a few years ago when mayor Boris Johnson pulled the plug on the funding, however, the plans were resurrected after Ealing Council found a plan to install a lift at the station for only £2m instead of the £10m price tag TfL had given it, and the council contributed £200,000 to the project.