That depends on the two institutions rather than on any processing schedule. Academic and research entities often route letters through separate administrative offices, so allow time for drafting, internal approval, and correction before anything is treated as final. Sequence follows responsibility: nothing is final before the party who must sign has actually approved it. Ask each organization to name the individual holding signing authority and the approval step above them. A schedule built on a coordinator's estimate rather than on the signatory's calendar tends to slip at the last stage.
Work backward from the earliest commitment
Identify which portion of the work starts first and treat that date as the constraint. Ask each organization who signs, what internal approvals a duty letter needs, and how long reissuing a corrected letter takes. Grant cycles and term dates can make a laboratory start date harder to move than an academic one.
Avoid resigning from current employment until the documentation is settled. Order the steps so each one is worth taking. Confirm the legal employer before commissioning a credential evaluation; settle the duties before requesting a letter; hold the letter until both agreements agree with it.
Should the professional also hold an ownership interest in a related U.S. entity, resolve that first, because self-employment is not permitted and a restructuring cannot be improvised at a port of entry. Status is temporary in every case, so record the intended end date while the sequence is still being planned.