While it's clear that you meant only to be polite and effective, it seems to me that there are some subtle issues with the emails you sent that could cause someone to respond negatively.
I apologize for emailing again ...
This seems a little odd. If it is your job or natural action to email again, then apologizing is a claim that your action is an imposition. Unless you are going to do a full "accusation audit" along the lines of "you're probably going to think I'm being totally unreasonable for asking this. [Message]" then I recommend leaving this out.
... but ...
Any apology that is followed by "but" is greatly weakened or even no longer treated as an apology. This single word is likely to remove any softening that the apology itself may have achieved. Consider that an apology acknowledges wrong done. Saying "but" after that lessens that acknowledgment because it justifies the very actions that one is apologizing for. If your actions are truly just, do not apologize for them. Saying, "I'm sorry, but [strong statement]" in certain tones is in fact a colloquialism to emphatically express that one is not personally sorry at all except in the sense of being sorry that the other person is wrong. Email has no tone of voice, and it is very easy for someone to get this alternate meaning.
... I have not received a reply ...
This statement that you haven't received an email communicates the subtext that the person has failed to meet your expectations. It is not a request, nor a call to action. It is not a neutral statement of fact—a more neutral statement would be "I haven't seen an email come through on this." While those may sound identical in meaning, they are very different in their connotations. "Seeing an email" allows for the possibility that you missed one that did, in fact, get sent. "Have not received a reply" is to call someone to task as if you were the boss of that person and you are discussing performance expectations.
... to my email beneath.
While this may seem to be neutrally stating the location of the email content, by using "my email" it makes it more demanding through self-focus, along the lines of "YOU didn't reply to MY email." This is a very different communication from "I'm following up on the below email." It is also more common to say "below" for referring to the location of content in an email. "Beneath" is fine, but to me potentially risks sounding like more formal speech, and formal speech is considered less warm.
I will appreciate hearing from you.
"I will" is not idiomatic in English. "I would" is the correct way to say it. By stating "I will", it comes across as a statement of fact about the future, that indeed the recipient WILL send you an email (and thus you WILL hear from them), and you intend for them to know you require this outcome. By saying "I would", you communicate the extra softening to it, "If you are able to respond to my request, that will result in me feeling appreciation." It is conditional. This is very different from the communicated subtext, "When you respond to my demand, I will feel satisfied."
Please let me know if you require more time.
Even with the "please" there, this feels like calling the recipient to task instead of simply asking for what you need. A manager speaking to his employee might reasonably define exact parameters of acceptable behavior. In this case, you probably don't have that kind of relationship with the counterparty. Is hearing about a timeline for a proper response really that much more useful to you than simply getting the proper response itself? Someone who is professional will communicate this to you on their own, and I would be hesitant to expect anyone to respond to this positively unless there is an implicit agreement and understanding that you possess the authority to demand compliance.
Here's what I would suggest instead. It's hard to get specific because I'm not seeing much detail about what the relationship between you and the counterparty actually is, or what task you are asking them to perform. But here are a couple of samples:
When you are working on project X and your own success derives from their tardy deliverable Z
My manager is asking me for an estimated completion date for my project X. The project is now waiting for the Z information from you. Is there anything stopping you from sharing an estimated timeline for completion of Z so I can let my boss know what to expect?
This removes all blame, all "calling to task" or strong criticism, and simply drives home the stark business details: you need Z so you can complete X. By using third-party authority (your boss), you are asking for help with a personal problem. This arouses sympathy, and makes the request easier to not take personally. By asking if anything is stopping the recipient from sharing a timeline, you invite him to say "no", which people like to do. You also are implicitly asking him to explain himself if there is a reason he cannot complete the task or cannot give you an estimate, but doing it in a very polite and non-confrontational way.
When you have the right to expect the other person to perform the task, such as to complete work you ordered
Just checking in on the status of the below request. Maybe I've missed a reply or something has come up on your end.
If you could share where things are, I'd appreciate it!
Why Be Soft
This may seem very non-formal, but that is quite intentional. One thing an email like this will rescue you from is your own human error or lack of information. Here are a bunch of possibilities where a soft email like this results in you looking just fine, instead of looking like a jerk:
The recipient did reply but your email system is broken. Or the reply-to address was wrong. Or you accidentally deleted the email. Or the email is waiting in your in-box but your email client is silently not communicating with the server and needs to be restarted.
The recipient doesn't work for the company any more.
The recipient believes correctly that the thing you're asking for is not his responsibility, and it is your company's records that are inaccurate in some way.
The recipient replied back to someone else at your company and the issue is already taken care of.
The recipient had a death in the family and has just returned to work.
The recipient's email server crashed and lost days of email with no way to recover it or even know that you sent the email the first time.
Your boss made a special exception for that recipient, without telling you.
... and so on and so forth.
In my opinion, it's always wisest to approach these things from the perspective of "hmmm, could you help me understand more about this situation" than it is to call people to task—even if one is the boss and ostensibly has the authority to do so.
One More Idea
It seems like the below email from a few weeks ago may have gotten missed. Would you mind taking a look?
In my opinion, if the issue is truly one where the counterparty owes you and is failing, this is one of the most professional and effective ways to address it.