The implementation timeline varies and is typically defined in the directive itself. After that, infringement proceedings are at the Commission's discretion. Several months are in any case required for translation, review, etc. and a full case, all the way to an ECJ decision, takes several years.
Fines are only possible once the ECJ has found a member state to be non-compliant, i.e. the infringement proceedings went to a conclusion and nothing happened after that. The Commission can then go to the court again, this time under article 260 of the Treaty on the Functioning of the European Union, and ask the Court to impose a fine.
But in sensitive domains, the Commission has to pick its battles and might choose not to act or to overlook some details. There are also domains where it did not historically have enforcement powers. That means that infringements can go on for a really long time in some cases. There is therefore no hard deadline or predefined delays, it depends mostly on the Commission's priorities.
As an example, consider the McCarthy decision rendered by the Court at the end of 2014. Without getting into the details, it stems quite directly from the wording of the directive (in my opinion) and yet it took more than ten years for the UK to actually recognise this and act accordingly (and that case did not stem from infringement proceedings).
States can also ignore directives for a really long time, either for political reasons or out of a genuine inability to act. This might involve doing something (but not enough), paying fines, etc. while still falling short of the original intent. For example, France is being famously difficult with the Birds directive. The Commission recently started formal infringement proceedings based on the 2009 directive (7 years after it was enacted) but the problems date back to the first directive in the 1970s.
(Formally, if you read the treaties or secondary literature, you might notice that member states also have the right to initiate infringement proceedings against another member state. But that has only happened something like three times in the whole history of the European Union. So in practice, it all comes down to the Commission's priorities and actions.)