email speed

From what you're saying, the problem is not the email sending speed. The problem probably comes from your payment gateway which only sent the payment notification half an hour after the payment was made. Unfortunately, there is not much you can do besides asking your payment gateway to send the payment notifications faster or use another payment gateway.

With the bank transfer plugin there is indeed no payment notification so it's a different problem than the one I explained above.
From your previous message I thought that the account registration email came immediately and that you were getting the order confirmation half an hour after.
But since now you're saying that your account registration is also long to get to your inbox, it's likely to be a problem with your email server.

I see that you're using the sendmail function in order to send your emails. That means that you're using your hosting company email server which is pre configured on your server.
You should ask your hosting company about that problem. They should be able to help you by either transfering you to another email server or by proposing you a faster way to send the emails. Also, you have the option of sending your emails via an external SMTP server like the one provided with gmail accounts so that could be also a solution.

smtp, host and sendmail are only connection methods to your email server. The connection can vary from 1 to 5 seconds maximum (smtp is a little bit slower), but that won't create gaps of 30minutes. That parameter is only important when you send thousands of emails in short periods of time with newsletters as the connection itself needs to be fast.

In your case, what slows down the sending of the email is the email server queue. That's why I encouraged you to contact your hosting company. They are the one controling the email server and who will know why the queue of the email server takes so much time to be processed.

For information, on our servers, we use sendmail.