It is traditionally for staff to request such letters when there is a change in management, so there is a good chance that there is more than a dozen letters to be written and approved, some difficult and possibly needing legal approval as well as HR. So I would not be surprised if ir takes a few months.
If your have been good in the past and you have no reason to believe that reference would be otherwise, why create a hassle. On the other hand if you expect it will be problematic, hassling won’t improve it - it will turn into a statement of employment written by HR and quickly signed by the manager to get you of their back.
Why not ask your former manager what the status is?
Interim letters are common even if your manager does not change, I have requested them multiple times in various roles and they are especially useful when looking at applying for new jobs as they give you the ability to present a current reference and indication of performance to a prospective new employer.
Depending on how good the working relationship was, you might take roegner's advice and complete the whole reference yourself. Alternatively, once you've done the above part of the that work for them (and it is in your interests to do so), you might find they're more likely to get on with it and add anything they think worthy of mention, especially praise, and ideally that they would gladly work with you again.
When someone is switching roles, got say 10 or 12 such letter to write and you are hassling for your interim letter for no good reason, you'll get something but it probably won't be as good as it might have been had the manager been able to take their time. Most managers take this serious and will do a good job on it if given the time.