If the police have an email, sent by a suspect over a 3G or 4G network, could they use the IP address (since they know when it was sent) to find out - from the service provider - the precise location the email was sent from?
Yes, this is very easy. However... the key word here is "precise location." Not exactly. Not unless the phone is hacked.
If you're looking for evidence of governments assisting law enforcement with locating devices, then you'd be looking for the NSA's Treasure Map program. This is available to cleared law enforcement personnel, mostly FBI/DEA, but I wouldn't be surprised if they also assist local law enforcement.
The NSA shares intelligence data with local law enforcement and helps them utilize parallel construction to make their cases.
ISP & Normal Law Enforcement options
Schroeder covered this pretty well, but let me add to it:
Since you're writing for TV, I feel you should know this part to make it seem more realistic. Anyone can walk into Walmart and buy a throwaway smartphone or dumbphone. From there, they can go to the nearest open wifi, and register under fake credentials. Fake name, fake address, fake everything else. And they can use a prepaid credit card that they purchased with cash to register the device(s).
So you won't be able to find their actual address, or even know who they are, unless you hack the phone (normally a smartphone).
However, if you know the general time-frame that someone bought and created the account, you can request evidence from Walmart, and they're usually almost always happy to help law enforcement. They'll be able to review the security footage to see who bought that device, and when.
But how will they find the time frame? Walmart, and other major retailers, keep track of when things are sold, right down to the very minute. You know when you return an item? They know, because the information is stored in their databases, and looking up the bar code of the receipt is possible. It shows when the purchases happened.
Doing a bit of investigation will probably reveal that the account for that phone was registered at a specific time. If the phone was registered at a specific time, then it may be likely that the perp purchased that phone at a nearby store.
Bringing up a list of stores in close proximity to the open wifi where you registered the phone may reveal where the perp purchased the device. You can then go in and request security footage to look for anyone purchasing the phone(s) in the electronics departments. Better yet, the place with open Wi-Fi may have you on camera at the time you registered.
Other Perp-Locating Options
And then there's Stingray, an IMSI-Catcher.
Since you know the perp's IP, you can likely find the perp's carrier. With the perp's carrier providing the phone number used by that IP address on their network, bringing up your actual cell phone number is not hard. In fact, if you know of an area that the perp has hung out at, you can use a Stingray device to perform a man-in-the-middle attack on the suspect without him realizing it.
Every mobile phone has the requirement to optimize the reception. If there is more than one base station of the subscribed network operator accessible, it will always choose the one with the strongest signal. An IMSI-catcher masquerades as a base station and causes every mobile phone of the simulated network operator within a defined radius to log in. With the help of a special identity request, it is able to force the transmission of the IMSI.
An IMSI catcher is an incredibly easy-to-use, one-button-fatality-man-in-the-Middle-attack-in-a-box. It allows law enforcement and intelligence agencies to act as a tower to catch communications. Having personally seen one in use, I can attest to their effectiveness.
Using normal tools, even those that don't require the help of the NSA, providers can generally help you find the location of any given phone at any given time. It knows the closest tower you're connected to at that time.
If you're able to force the location feature to turn on, which law enforcement can do... how do you think 911 finds you when you can't tell them where you are because you don't know? They can know the general area you're at, within a few hundred feet.
IP Address Geo-Location in USA and China. NEVER rely on this!
While, yes, it's certainly possible to geolocate a phone's IP address, you should not rely on this because the information returned can be wildly incorrect. Your assigned IP address, even if you're somewhere else at the moment, could be shown as elsewhere.
In fact, when I travel all over the place, and tried to geolocate my IP address, it was always located in the city I registered in. I've tested this both in China, and in the USA. I could be 2000 miles away, but the phone's IP address geolocates to a different state/province.