I think what they are going to serve is a PIN, which is a "Police Information Notice". I've come across these in my profession, where they tend to be used tactically by ex partners.
The PIN is an informal way the police have developed of dealing with low level or unproven harassment claims. They are obliged to prioritise stalking/harassment complaints, but many are low level and borderline and rather than do nothing, the other option is to warn the accused (and this is what a PIN is for). Most have the desired result: the complained of behaviour stops. For the ones that don't, they are then used as evidence in any prosecution that the accused was put on notice of the complaint, was warned to stop, and was made aware of the criminal offences under the Protection from Harassment Act.
The PIN is meant to be a simple warning. It has no official status (it is not anything like a caution). Due to its lack of status it should theoretically be pretty meaningless, however it isn't and it stays on someone's record for a year. The reason it is "meaningless" is because it is served without any findings or real investigation. However, it goes on the national and local police data bases and most people assume it is a bit like a caution and it scares them into backing off.
The PIN will record that a complaint has been made, what the complaint is and it explains the offences under the Protection from Harassment Act, and that if the behaviour continues then they will be committing a criminal offence and they may be prosecuted. Where the harassment continues, it's used as evidence to avoid the offender claiming they didn't know it was harassment.
Most PINs have the desired effect of nipping these sorts of things in the bud.
In your case I'm not sure it will have any effect, because the police have already been involved and it's done nothing.
Once a PIN is served it can only be withdrawn if an official police complaint is made, and successful complaints are very rare.
I actually think that your case is worthy of much, much more, but if it's what the police have decided there isn't much you can do other than carry on evidence gathering and reporting further acts of harassment.
Google PINs and the Protection from Harassment Act - there's an awful lot of information out there for you to fully educate yourself. Having read what there is, you may feel that this is something you can push further with the police.
I think a caution would have been much better. Something much more formal and scary than a PIN. But they may have already had advice from the CPS that there is currently insufficient evidence to prosecute (and therefore the PIN may be the only other option - although they could presumably have given a caution, but of course the accused has to agree to accept the caution (I think)).