HSA and Medicare

You are generally correct that once you are enrolled in Medicare, you are "no longer eligible to make contributions" to you HSA.  However, this is really an issue of how much you can contribute to your account as opposed to a timing issue.  If you are eligible to contribute any amount to your HSA for a given calendar year, you have until April 15 of the following calendar year to make your contribution, even if you are no longer "eligible to contribute".

In situations like you describe, HSA contribution limits are prorated monthly.   HSA contribution eligibility is determined as of the first of each calendar month.  For example, the 2024 individual HSA annual contribution limit is $4,150 and individuals over age 55 can contribute and additional $1,000 per year (the catch-up contribution).   If you will be enrolled in Medicare for any months of 2024, you should divide the $5,150 combined annual limit by 12 months to come up with a $429.16 monthly contribution limit.  However many months you are covered under an HDHP but not Medicare would become your 2024 contribution limit.  If you were not enrolled in Medicare until June 1, you would have five months of eligibility and could contribute $2145.83 to your HSA any time before April 15, 2025.

The first part of your question is impacted by Medicare's retroactive enrollment rules.  When you enroll in Medicare after you turn 65, Part A enrollment is retroactive for up to six months (or back to your 65th birthday, whichever is more recent).  That retroactive enrollment in Part A is what makes you "no longer eligible to make contributions" for any months of retroactive coverage.  Essentially, you would not count any of those months towards your $429.16 per month 2024 HSA contribution limit.