After removing the message all the parties in the conversation will see the "removed this message" where the original message used to be as shown in the screenshot here.
Edited to add: If the user that is supposed to receive the message has their unread messages notification enabled they might still receive the message that was subsequently deleted. It depends on the time period for the notification that the user selected in their settings.
re: "does the client will see my original chat via email?"
TECHNICALLY: Depending on her notification settings, if the client has the Upwork Messenger tool up and active, then a message may NOT be sent via email, and will only be shown on screen. If the client has the Upwork Messenger tool active, and is not looking at it, then she may NOT see the original message show up in the tool before it disappears, AND she may never receive anything via email.
So it is POSSIBLE that a client will not see a message you send before you delete it. But those are some significant "if"s.
So you may as well assume: "Yes, the client sees your original email message."