You can -- and should -- immediately appeal an order that effectively dismisses class claims but preserves individual claims. The order is tantamount to a final judgment for the putative class members. And it discourages the named plaintiff(s) from proceeding to an appealable final judgment. The only effective avenue for review in an immediate appeal.
You cannot -- and therefore need not -- immediate appeal an order sustaining a demurrer to claim claims AND individual claims. Final judgment is coming soon enough. You can wait until then to appeal. And that appeal will not be dismissed as untimely.
The Supreme Court explains today in In re Baycol Cases I and II:
"The right to appeal in California is generally governed by the 'one final judgment' rule, under which most interlocutory orders are not appealable. (See Code Civ. Proc., § 904.1.) In Daar v. Yellow Cab Co. (1967) 67 Cal.2d 695 (Daar), however, concerned that orders dismissing all class action claims might in some instances escape review, we adopted a 'death knell' doctrine that allowed a party to appeal such orders immediately. Here, we address uncertainty over the scope of the death knell doctrine: does it extend to orders that simultaneously terminate individual claims as well, or does it apply only where, as was the case in Daar itself, individual claims survive?
"We conclude the preservation of individual claims is an essential prerequisite to application of the death knell doctrine: the doctrine renders appealable only those orders that effectively terminate class claims but permit individual claims to continue. When instead an order terminates both class and individual claims, there is no need to apply any special exception to the usual one final judgment rule to ensure appellate review of class claims. Instead, routine application of that rule suffices to ensure review while also avoiding a multiplicity of appeals."