They did not ignore you because they are cruel. They ignored you because the last ten messages that said "urgent" were not. Attention is rationed against a history of theatre. Most urgency is a lie. The rare message that tells the truth gets answered because it is usable. It contains a day, a reason, and a next step that does not require telepathy.
Honesty is not a tone. It is a set of facts a tired person can plan around.
A real day beats a feeling
"ASAP" is your anxiety. "Wednesday 5pm Eastern so we can tell the vendor Thursday" is a constraint. The second one can be put on a calendar. The first one can only be felt, and feelings from other people's mail are the first thing people mute.
Respect their time by making the deadline checkable. If you do not know why the day is real, you do not have urgency. You have a preference. Say the preference. People will often meet a preference. They will not meet a fake fire twice.
A default is a kindness
The reply is the point. If silence leaves you frozen, you did not finish the ask. "I will proceed with B on Friday if I have not heard" gives them two ways to answer: pick something else, or let the default stand. Both close the loop.
That sentence feels blunt. It is blunter than a third "just circling back," and it spends less of their goodwill. A default is not a threat if the consequence is something you were going to have to do anyway.
If you will not proceed without them, say that too — and then you need a meeting or a later day, not a louder adjective.
Theatre trains silence
Every fake ASAP teaches the reader to wait for the version that contains a date. After a while they wait so long that even your real deadline looks like costume. You cannot fix that in the subject line of this one message. You fix it by sending fewer fires, and by letting a missed fake deadline pass without a scolding.
Scanners are not reads. Do not escalate because a link was fetched. Do not write "I can see you opened this." You cannot, honestly, and the accusation spends the last of the trust you needed for the actual date.
Write it so a Monday reader can act
They may open this after a weekend, with no AI, on a phone. The TL;DR has to carry the day and the ask. The note can carry the human part: "I am sorry this is tight. Friday is the vendor's cutoff, not mine." That is urgency with an owner.
Approve that. If the assistant wrote "I need this ASAP!!" delete it. You are not sending a feeling. You are sending a day they can meet — or a default they can live with.
People answer honest urgency because it is rare, and because it is possible to answer. Give them a job they can finish. They will finish more of those than of your alarms.