Hey yeah this is the process I went through myself..
Called Apple about another issue with new Mac mini and brought up the other Mac and asked for advice. He opened another service ticket and advised me to request assessment under EU consumer law claims. As there is no Apple Store near me I used an AASP (authorised Apple service provider). I think this may have worked in my favour as it's more independent if u ask me.. I'd advise to use an AASP instead of going straight to Apple. They can tell you where the closest one is when you call.
I dropped the Mac mini off to the AASP and filled in the relevant legal paperwork. (This required original receipt and photo ID)
The ball is now in their court for up to 2 weeks. Engineer looks at the Mac, fills in the paperwork and if he thinks it's possible then will submit a claim to Apple for the relevant part (logic board) and wait for Apple to either Approve or Deny the claim, get the part sent out to the AASP then hopefully a happy customer!!
One thing I did think I could have done after I had dropped the Mac off was added a small note saying about the issues, how it's quite common looking at this thread and how it can be fixed by cooking the logic board as suggested here. Just adding anything to support the claim..
As I'm quite impatient I was calling the AASP daily for updates as I'd still be in the dark about how it was progressing if I hadn't.. anyway, if there is any other information you require from me feel free to ask and I will reply at my earliest opportunity..
GOOD LUCK :-)
Stuart