Bute are lottery for 4+, that happened a long time ago for 4+ Sep 2023 entry, I mean people accepted offers in March 2022! So all offers went out last year . You might be able to go on waitlist, but it would be pointless as so many mums desperate to send their DD's are on there from the original lottery, and we're not talking about a lot of places in the first place.
All the other 'Bute reject' girls prep will be booked up, but you never know if a place might pop up, your playing catch up. Just contact them, they are very approachable, but I suspect your only hope is 7+ entry now, where they tend to select the brightest girls as the 4+ is mixed entry.
Best thing would be to speak to Hammersmith and Fulham council about a state school place so your DD can at least start somewhere, just so you know pretty much all London primaries are of a good standard especially as so many middle class parents choose them because they can no longer afford Prep and are saving for private senior school, waiting out for grammar school.
While you do that, you'll just have to blanket approach all girls preps and if I were you apply to the all the all through girls schools, most of them will want to assess your DD before offering, all of them will have sent offers and have waitlists, BUT this is London, parents lose thier jobs, partners drop dead ( don't mean to be heartless, but it happens) Jobs get moved to New York, etc, so there's always movements - the trouble is you'll probably have to take what's on offer.
'Worst' case is your DD attends a decent enough state primary and a place comes up at a decent Prep or Junior dept of an all through school by year after Reception ( Year 1) You could just settle at state primary allocated ( they are obliged to find somewhere for your child) as long as its fine enough, and go for 7+ at Bute or any private school, when things open up again.
Don't worry too much, years 4-7, be it Bute or local Primary all follow basic same thinig, Early Years Foundation. Mine are at Prep and get broadly the same as their state primary cousins, except mine get a specialist French teacher, Music teacher, smaller classes, better facilities etc, better sport and after school clubs, more school visits and trips. Yes these things are important, but you can make up these things with private clubs and afterschool clubs with the small fortune you're saving in fees.