To add to what ccieexpert noted, it would also be valuable to understand what you consider a delay.
Are we talking about minutes, or hours?
(minutes is no uncommon, since email isn't real time communication, and for emails with attachments up to an hour can be expected in some edge-cases.)
Regarding "how to troubleshoot", one of the ways would be to look at the email header on an inbound email that you received.
In outlook you can, for example, just open up the email, go into properties and look at "Internet headers".
The email will include an "Received" header for each SMTP gateway it goes through, (unless it's deliberately removed).
The Received header will include timestamps.
For example, consider the headers from this email I got in 2021, and tried to remove identifying info from.
Received: from <my email srv> (LHLO <my email srv>) (<my email srv ip>) by
<my email srv> with LMTP; Tue, 26 Oct 2021 12:58:58 +0000 (UTC)
Received: from localhost (localhost.localdomain [127.0.0.1])
by <my email srv> (Postfix) with ESMTP id 4AB181105CAB2
for <my email>; Tue, 26 Oct 2021 12:58:58 +0000 (UTC)
Received: from <my email srv> ([127.0.0.1])
by localhost (<my email srv> [127.0.0.1]) (amavisd-new, port 10032)
with ESMTP id ID2ayBhvnpD4 for <my email>;
Tue, 26 Oct 2021 12:58:56 +0000 (UTC)
Received: from localhost (localhost.localdomain [127.0.0.1])
by <my email srv> (Postfix) with ESMTP id 79E171105CAB3
for <my email>; Tue, 26 Oct 2021 12:58:56 +0000 (UTC)
Received: from <my email server> ([127.0.0.1])
by localhost (<my email srv> [127.0.0.1]) (amavisd-new, port 10026)
with ESMTP id yFd9eV9UOJoR for <my email>;
Tue, 26 Oct 2021 12:58:56 +0000 (UTC)
Received: from <my mx hostname> (unknown [<my mx ip>])
by <my email server> (Postfix) with ESMTPS id 33CEB1105CAB2
for <my email>; Tue, 26 Oct 2021 12:58:56 +0000 (UTC)
Received: from <remote MX hostname> ([<remote mx IP>])
by <my mx hostname> with ESMTP/TLS/ECDHE-RSA-AES256-GCM-SHA384;
26 Oct 2021 12:58:34 +0000
Received: from unknown (HELO <remote exchange hostname>) ([<remote mail srv IP addr>])
by <remote MX hostname> with ESMTP; 26 Oct 2021 12:58:34 +0000
Received: from <remote mail srv hostname> ([::1]) by <remote mail srv hostname> ([::1]) with mapi id
14.03.0513.000; Tue, 26 Oct 2021 12:58:34 +0000
You can see that the remote email server (exchange) sent the email at 12:58:34 to it's local MX, and over to my MX, and there's a 22 second delay before my MX sends the email over to my email server, and we can see another 2 second delay in internal processing between components.
This is what you could to to identify where the supposed delay is happening.
If you're operating your own mail server & gateway, logs from those systems is what you could also be looking at.
