Charles didn't have to be "named king." There's no official ceremony for becoming king, there's no oath of office; there's no swearing in ceremony; there's no document to be signed. Charles simply became King the moment Elizabeth died. That's how it works.
King Edward VIII was in fact King Edward VIII even though he was never "crowned."
Actually there is.
Yes, the title of King immediately transferred to him on the Queens death he will not be officially proclaimed King until announced by the Accession Council which will happen at 10am UK time on Saturday.
This happens in two parts. The first is a meeting of the Privy Council without the monarch present to proclaim the sovereignty and formally approve various consequential orders.
The second is the new monarchs first meeting with their Privy Council where he will make his Declaration and read and sign his oaths and approve various Orders in Council to facilitate the continuity of government.
Traditionally the whole Privy Council would be summoned for the event but as there are now over 700 of them that is not practical. Still, 200 of them will be there.