Begin with the two slowest steps, the state licence and the health-care certification, which are measured in months. The port-of-entry application itself takes a day, but only once those exist.
Work backwards from the slowest third party
The mistake is to plan from the hospital's preferred start date forward. Plan instead from the licensing board's written processing estimate and the certification body's timeline, whichever is longer, and add a margin for a request for further documents. Only then fix the start date with the hospital, and ask it to date the employer letter close to the planned entry so the period requested is current.
If the hospital insists on an earlier date, the alternatives are a delayed start or a different unit of work that does not require the pending licence, not a border trip on hope. Once admitted, record the TN expiry and the licence and certification expiries side by side; the renewal sequence should begin several months before the earliest of them so the second entry does not repeat the first delay.