Tracy Brabin, who made public transport her key election pledge last year, has written to Arriva to complain about the move, which will see four local services axed by the end of February.
Ms Brabin took her first steps towards bringing the region's buses under public control in June last year, but the process is unlikely to be finished until 2025 because of government-enforced timescales.
The campaign group Better Buses for West Yorkshire called for "faster action" on public franchising, while local Liberal Democrat councillor Tom Gordon claimed recent events suggested Ms Brabin is "not fit for office".
The mayor herself has said she is "extremely frustrated" by the length of time public franchising is likely to take.
Ossett and the Five Towns are among the areas most heavily impacted by Arriva's decision, with the 117, 119, 184 and 187 buses all set to be withdrawn.
A raft of other timetable changes will also be made, with many services becoming less frequent and entire communities being cut out of routes.
Arriva says the alterations have been fuelled by low passenger numbers in some areas.
But Matthew Topham, from Better Buses for West Yorkshire said: "These axed routes and drops in frequency, motivated as companies focus on paying dividends rather than supporting passengers, will put additional strain on already undeserved communities.
"There will be more missed hospital appointments, fewer job opportunities and more expensive, multi-leg journeys — just as the cost of living spirals."
Mr Topham claimed public control of buses would save both services and "millions of pounds", adding: "We need faster action to deliver that change."
He added: "Public control would allow cross-subsidy where the profitable routes, like those run by First in the middle of Leeds, are used to pay for quieter but essential routes around Wakefield. It's time passengers, not shareholders, came first on Wakefield's buses."
Councillor Gordon, who represents the town of Knottingley on Wakefield Council, said he was particularly concerned about the loss of the 188 service to the town.
The service previously connected Knottingley with Castleford and Wakefield, but from the end of February it will only run between Castleford and Ferrybridge and be renumbered the 158.
Coun Gordon said: "Having to get two buses between Knottingley and Castleford when it’s a route vital to people’s livelihoods isn’t feasible.
"In Knottingley, where’s the incentive for people not to get in their cars, if indeed they can afford a car in the first place?
“Tracy Brabin made public transport her number one pledge when she was elected. She can’t pass the buck on this.
"If she’s not delivering on this, is she right for the job?
"All we've had so far is a token gesture people can get on the buses for free one day a week.
"She's not fit for office."
In response to the criticism, Mayor Brabin said she'd "consistently opposed" any proposed cuts to bus services in the region.
She added: "I’ve written to Arriva UK’s chief executive, to raise objections to the cutting of these important bus services in Wakefield and Kirklees.
"One of my pledges as Mayor is to gain greater local control of bus services, so they can be improved for everyone’s benefit and better connect people to jobs, education and opportunities.
"We have already committed close to £1 million to do this. But our work is being held back by the need to work to the government’s timescales.
"Both my fellow leaders in the region and myself are extremely frustrated by this."
Mayor Brabin said her improvement plan for the region's buses set out ambitions for simpler fares, contactless ticketing, greener vehicles and more reliable buses.
She added: "We’re already in discussions about an enhanced partnership with the region’s bus operators to progress this.”
Local Democracy Reporting Service