As reported by British Comedy Guide, the fifth season of "Jack Whitehall: Travels with My Father" will premiere "later in 2021," which is not very specific. Netflix renewed "Travels with My Father" for a fifth season on March 11, 2021. In its renewal announcement, Netflix also stated that filming for the fifth season would begin in April 2021, which would mean production is currently well underway.
We can make a guess for when Season 5 will premiere based on the timeline for Season 4, which saw Jack and Michael travel Down Under. That season filmed during February 2020, at the same time as Jack's stand-up comedy tour in Australia and New Zealand (per Chortle). It premiered in September 2020.
But this season probably won't need as much time between the end of filming and the premiere. For one thing, there (hopefully) won't be the arrival of another global pandemic that will throw TV productions into disarray. Like many other TV shows, "Travels with My Father" now has a year's worth of experience operating within COVID-19 protocols. Season 5 also likely won't need as much time because it takes place much closer to home.