Gordon Brown is the "next" PM not to have the Garter, Tony Blair, his predecessor was made a Knight of the Garter in 2022. However, Brown was made a member of the Companions of Honour in 2024 which I think is a bit too soon for him to get the Garter this year. It has also been rumoured for a long time that Brown doesn't want a knighthood and often people who don't want a knighthood are instead offered to be a Companion of Honour as it brings not titles with it but it still relatively exclusive.
If Brown isn't one for getting the Garter it might prove problems for those who have come after him as it would potentially look like the King was favouring one party or former Prime Minister over the other. But the next to come after Brown would be David Cameron, now Lord Cameron after he was elevated to the Lords to serve as Foreign Secretary.
Gus O'Donnell, former Cabinet Secretary (then highest official in the Civil Service) 2005-2011 including the 2010 election which saw the first hung parliament since 1974 is another possible contender maybe.
I suspect thought that Charles will want to pick some people who are more representative of the wide range of backgrounds in the UK rather than just white male politicians and senior civil servants.
I'd love to see Tim get the Garter - and he is "next", if you go by length of time married, after the Duchess of Gloucester got hers last year. However he is not a HRH nor does he technically carry out official duties other than accompanying Anne - his aren't listed individually in the Court Circular.