I don't know very much about the exact process, but the ECHR is not really an appeal court above our own Supreme Court. So, it would become a case by (probably) Archie Battersbee v United Kingdom. The argument in court would essentially be that the United Kingdom has infringed on Archie Battersbee's human rights through the way in which its law is written and/or its court processes have been applied.
ECHR case law on withdrawing life sustaining treatment is considered to be fairly settled since a French case (Lambert) in 2015. There is essentially a 3 part test that a country must satisfy.
(1) is there a regulatory and legal framework in place which is compatible with Article 2 of the Convention (the bit that gives a right to life)
(2) have the previously expressed wishes of the person concerned (ie Archie) been taken into account in the legal process, as well as medical opinions
(3) can the decision be appealed in the courts if there is any doubt about the best decision
The court also gives quite a lot of latitude to individual countries in applying the convention in these circumstances, as there are different approaches to end of life care in different European countries.
Given that the fundamental system used in England and Wales hasn't changed since either the Charlie Gard and Alfie Evans cases, parts (1) and (3) are essentially already passed by the UK. The only question would be about (2) taking into account Archie's wishes. And, the High Court and Supreme Court judgements are all clear that they have explicitly taken into account reports of Archie's previously expressed wishes.
Of course, it's possible that they may try a novel or unusual argument, based on a different part of the convention, or applying it differently.
However, most of the legal people I've seen commenting on this believe that the reason that this hasn't gone to the ECHR earlier in the process is that there is no realistic prospect of success.
In terms of timeline, in Charlie Gard's case 9 days passed between the Supreme Court decision and the ECHR decision. But there was an actual issue of law to resolve, as that involved an alternative treatment possibility, which so far hasn't been proposed in Archie Battersbee's case.