The term 'criminal proceedings' covers a multitude of circumstances and not just charging a person with an offence.
There are other ways to instigate the commencement of criminal proceedings without a formal charging process. For example, a summons is another method of bringing a person before a court - the person would never have been 'charged' with the offence(s) [albeit the matter(s) is/are put to the person by the court at first appearance]
In the same way as a summons is applied for, an arrest warrant also needs an 'information' to be laid before the court.
The court then decides whether to grant the summons or warrant based on that 'information.'
The laying of such an 'information' counts as the commencement of criminal proceedings in the strictest sense.
The normal process for a European Arrest Warrant [in this case an arrest - not a conviction EAW] is somewhat the same. However, the police will always seek the earliest advice from The CPS who will direct their actions.
As a general rule, The CPS will ask the police to submit an advice file and The CPS will then direct the police to obtain a normal domestic (UK) warrant for the alleged crime. Once issued by a court (following the laying of the 'information' mentioned above) to the police, it is normally the role of The CPS to decide if an application for a European Arrest Warrant is justified and, if so, The CPS apply to the court (not necessarily the same on which issued the domestic warrant) for an EAW.
The court have to satisfy the criteria necessary for the issue of an EAW and, if satisfied, they issue one.
That EAW is then usually circulated in the normal way [via Interpol etc.]
All of this process can be described as the commencement of criminal proceedings.
Whilst the police appear to be the butt of the criticism here, the police would have acted on CPS advice and that, in itself, would have been based on the information supplied to the police by The NHS.
Normal procedure would then have been for The CPS to have directed police and for The CPS themselves to then apply to the court for the EAW.
The Judge/Magistrate must then have agreed to issue it.
FWIW, the police are between a rock and a hard place here. They are only responsible for collating the information (evidence) and, in this case, that will be wholly based on what The NHS have said. The police then present it to The CPS. Once done, the matters are for The CPS to decide.
Aside from the question as to whether this was a sledgehammer to crack a nut (that sledgehammer presumably having been wielded by The CPS - not the police), the key issue is if the police (and CPS) had done nothing and the child had died en route to Spain, what would have been said then?
All just IMHO of course.