As early as the plan allows. Institutional record offices work on their own schedules, and a closed or archived record from an earlier enrolment can take considerably longer to retrieve than a recent one from the conferring school.
Treat the older record as the long pole
The conferring institution's transcript is usually the quicker of the two, since the student is recent enough to sit in an active system. An enrolment from years earlier, at a school that has since changed systems or merged, may require an archival request. Start that one first and let the rest of the preparation proceed in parallel rather than waiting on it.
Companies changing their legal name are already reissuing contracts, payroll records and letterhead, so the corrected support letter costs little effort if it is requested during that work. Take the date on the admission record and count back far enough to gather a revised letter, obtain legal review, and complete whatever step applies. If credentialing requires verification from a medical school or a training programme abroad, that request should go out early, since it depends entirely on another institution's response time. Write out the proposed early activities hour by hour and ask whether they produce value for the U.S. employer, involve its clients or systems, or resemble the future paid duties.