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From Wikipedia,

Development began in 1996 for a launch that was initially planned for 2007 and a 500-million-dollar budget, but the project has had numerous delays and cost overruns, and underwent a major redesign in 2005. The JWST's construction was completed in late 2016, after which its extensive testing phase began. In March 2018, NASA further delayed the launch after the telescope's sunshield ripped during a practice deployment. Launch was delayed again in June 2018 following recommendations from an independent review board. Work on integration and testing of the telescope was suspended in March 2020 due to the COVID-19 pandemic, adding further delays. Work has resumed, but NASA announced that the launch date has once again been delayed to October 31, 2021.

Now,

NASA had been working toward an October 31st launch date for the James Webb Space Telescope, but it's having to delay the science observatory's trip into space once again. Thankfully, the launch might take place just a few weeks later, in November or early December. A rescheduled date is unlikely to be confirmed until later this summer or perhaps in the fall.

XKCD provided an extrapolated launch date of around 2026, given previous launch delays.

When will the James Webb telescope be launched?

This question resolves on the date during which the first component of the James Webb telescope is first launched upwards towards (whether or not the telescope functions as intended, or indeed even lasts more than one second after launch without blowing up), as described by reliable media.