Labour must be disappointed at only reaching similar levels to what they achieved under Corbyn. There's talk of Starmer having to resign if he gets a FPN and then the whole Left/Centre thing will start again and there just isn't time to go through all of that before the next election.
Labour also securing London while not making headway in the (real) country just reinforces that they are now the party of the affluent middle-class which must be worrying for them as that doesn't win elections - they will need a strong Midlands and Northern return at the General Election as they can't rely on the South-East outside of the capital.
For the Tories, it's a wake-up call but it's an expected wake-up call and it looks like they just about got away with it. Most of the Labour gains were from NOC and winning Harrow from Labour negates one of the so-called 'symbolic' London councils although what's symbolic about Barnet I have no idea.
Finally, it looks like the nationalist parties - Sinn Fein, SNP and Plaid Cymru - made gains which says to me these parties have a core 'shock troops' of voters who will always vote for them even if you dangle their cat over a canal. Those who are willing to change their vote - Tory to LibDem, Labour to Green - need to be wary of these people. They're the true nutters of British politics, not those voting for mainstream parties.