Well, if he was already a suspect, you wouldn't need the email to begin with. The investigators could have been watching their mobile phone wanderabouts the whole time (or another agency have already put this guy on watch, and thus the mobile has more data about it).
The other option is that you have an email, but no idea who the criminal is (for example, “They kidnapped my child and now I received this ransom email from [email protected] saying they are holding him in Eastasia…”).
Assuming the email was sent through SMTP and not by webmail, the IP address from which it was sent would be directly available to the investigators (show some Received: lines here).
Additionally, they could gather more information from the email provider (Google here), which could provide more information, in addition to other IP addresses from which he has connected, such as a phone number used for account recovery (if they have been dumb), the registration date (the day before, quite uninteresting), that the language used in the signup was German (this would be useful), maybe they even a Google Maps search for an isolated place that would be ideal for hiding someone (make them receive this when the guy is about to kill the poor boy)…
As stated before, geolocation is unreliable for determining where the suspect is (albeit immediate, so I would expect them to query it anyway), but it can be used to know where it isn't. If the IP address is geolocated to the city where the crime was committed, that means the criminal sent it from there, not from Eastasia! That was probably a bluff.
Once they have the IP address(es), they will ask the Internet provider (with a court order) who was using that address at that time. If it was accessed through 3G/4G, then they could ask for the location of such phone at the time of sending, and discover which tower service it (they also asked where it was now, but it's currently powered off).
However, it is also possible that he wasn't connecting through 3G, but through Wi-Fi (or that some of the multiples IP addresses they got from Gmail / several exchanged emails). Maybe it turns out to belong to Starbucks. They may then quite confidently assume -something they could check by connecting themselves from there- that it was sent from the only Starbucks premise in town (later they will find that the phone card was bought in a nearby supermarket). Or it may be a local coffee shop that happens to host their website on the same IP address used to nat the connections on their free Wi-Fi (not a good setup, but it was installed by the owner's nephew, and they only have an IP address). Thus, just entering the IP address in a browser they would learn the precise place from which it was sent. With no delays by legal roundtrips.
Knowing the store "from" which the email was sent may or may not be too useful. There could be interesting footage from security cameras. Perhaps he only went there once. Maybe he lives nearby, or even is able to connect from his home.
Naturally, if the criminal connects repeatedly from there, they can put it on surveillance, as well as immediately going there as soon as a new email is received.
