If there is truly a deadline that "if you don't give me the form by the 12th, your request can never proceed any further and what you have sent me can go in the garbage" then you have to let your correspondents know that, because it's really unusual. If the message is more like "I can't move forward on your request until you give me the form" then say that. Why give them a deadline? They're the ones who are in a hurry to get the thing done. Or not. Why would you care?
Also, it's generally considered polite to express a wish to be able to help the person but alas unfortunately be unable to because Reason, rather than just acknowledge they asked and then say Reason.
I would reword the sample above as
Dear Patient
I have received your request [for a hospital referral, laboratory test, treatment, etc.]. Unfortunately the signed form [or lab work, or updated Health Card, etc.] was not included. As soon as I receive that, I can proceed with your request.
Thank you for your attention.
Yours faithfully
Niece
or, in the unlikely case the request will expire,
Dear Patient
I have received your request [for a hospital referral, laboratory test, treatment, etc.]. Unfortunately the signed form [or lab work, or updated Health Card, etc.] was not included. As soon as I receive that, I can proceed with your request.
Important: if the signed form is not received by DATE, [the sample will be too old or whatever] and you will have to start a new request.
Thank you for your attention.
Yours faithfully
Niece
To summarize: don't give them a deadline unless it's a real deadline ("my boss will make me remind you in two days" is not a real deadline) and try to imply that it's unusual for them to have left out whatever they've left out, to encourage them to include it next time.
Your samples are generic, so this may be "off" but I would encourage you to look for process changes that can prevent this. The concepts of someone mailing you "please do x" and you saying "I need form y" and then your boss coming and saying "hey did you get form y yet, you should remind them" strikes me as inefficient. Couldn't the signed form be what kicks off the whole process, and without it there is no request and nothing to follow up on? Or perhaps person A asks for x, but you need form y from person B. If you are reminding person A about it, be more explicit -- not "I need form y" but "I need you to get person B to send me form y". If you are reminding person B, include something like "Person A requested this 4 days ago and we still need form y from you."