Right, firstly, don't worry.
Secondly, if matters are exactly as you have laid out there is absolutely nothing to worry about.
If and only if the police turn up, show them her emails to you. They may try and give you a harassment warning (which is basically a note you sign saying you will not harrass her in future). You probably ought to take legal advice on this, but my view is that you should not sign it.
If you have had enough of the police, unless they decide to arrest you, you can ask them to leave. They must do so unless they have sufficient to arrest you. On what you have said, I don' think they have sufficient.
Personally, I should try and relax and try not to worry (easy for me to say, I know, but as someone who gets the police called round here for less than farting, I know how reasonable they can be).
Personally, I'd leave it now and NOT WORRY. If the police were that fussed, they'd have been round. I suspect they are not fussed because there is naff all in in.
Your baby is more important now.