[csomar]

I don't think so. It's fair to ask for a timeline and even "slightly demand" from the author if they can speed up. (like "Can we please have this by ...). However, mentioning your clients as such (highly regulated industries) seems to me (I might be wrong) to imply a certain kind of coercion and responsibility for the author (hey look there are some very important people here using this.). It's not the responsibility of the author, and he shouldn't be stressed about it.

Of course the email seals the deal.

[brightlancer]

I think that mentioning the clients is providing Why the guy is asking; it provides context.

I don't read anything coercive in that. Desperate, maybe, but not coercive.

[csomar]

I don't see the point of such context. The client is not the responsibility of the OP.