Technically, UK has to initiate Article 50 proceedings. In real life, they can't sit around and wait at their convenience because they're the dog that chased the car and the car stopped without warning. They do not have the luxury of sitting back to contemplate the unexpected situation over the course of weeks or months.
What happens in real life is all the financial entities who account for 15% of UK GDP in the services sector want to know right away what a timeline will be. The EU wants to know right away, so France, Germany etc can plan what to tell their electorates in the elections due there within the next year or so. EU have no patience anymore, and the UK-EU engagement was always a tenuous one. Frankfurt and Paris want all the financial business London gets from their EU passport, which will soon be revoked. We're talking hundreds of billions, if not trillions, in money. What Merkel says and what they do are different matters. Clearly they want to ensure UK-German trade remains good. But France is not on the same bus, and will be more punitive.
They're not going to sit around patiently for 4 months while BoJo and co go "Blimey, we didn't quite expect to be in this position, mates. What now ?" over tea and crumpets.
Also, remember, the British establishment (and news) sources will deliberately state that 'lets do this slow and easy'. We need counterfactual news from the continent, rather than go by BBC alone. BBC and co are going to do their best to present a 'it's all under control. calm down, it'll take years' point of view.
The bottomline is that BoJo and co will face pressure from within and outside, and the timeline is not theirs to set. They're already squeezed into a corner they never expected to be in. Having already shown themselves to be incompetent idiots, they will not also be given the luxury of setting a timeline at their own leisure. The people who hold the $$ make the timelines, now that UK chose to shoot itself in the foot.
Longer wait = less favorable terms for UK. The whole argument for quitting was being able to negotiate as well as the Swiss have. It's not that simple. The Swiss could get good terms because UK for them. Now UK and Swiss are both out of EU, and EU has no incentive to offer either of them a patient ear.