guides for museums in London

I can’t suggest a private guide but I took the tour of the Tate Britain offered by London Walks (guide Rick Jones) and thought it was excellent. We had about a dozen in the group the day that I took the tour.
https://www.walks.com/our-walks/tate-britain/

I’ve also taken their tours of the British Museum and V&A pre-pandemic so don’t remember the guide but they were also excellent.

If you don’t want a group tour, they have a link for private tours on their website so you could inquire with them.

The British Museum and the V&A are monsters. It takes multiple days to see them fairly comprehensively. If you opt to pay for a private tour (and I doubt you'll find a different option for your half-day time frame), you should spend some time with the websites of those museums, identifying the parts of their collections you are especially interested in, then share that information with your guide.

I haven't taken the kind of private tour you're looking for, but from time to time someone mentions Context tours. That company seems to have a reputation for highly educated guides; your guide for the Tate (Modern or Britain?) will probably have a graduate degree in art history, for example. Here's what Context offers in London: Context tours in London.

For walking tours of historic neighborhoods or historic periods, I am a big fan of London Walks. Its advertised tours are public, but I think they can be arranged as private tours as well. The guides are licensed and engaging, and I'd recommend this less-cerebral option if you want a more casual, outdoor experience to balance your museum time.

I took the London Walks tour of the British Museum a few years ago. It was fine, but I don't think it was what you are looking for. Be aware that conditions at the British Museum--not the fault of London Walks or any other tour company--are typically extremely crowded, which isn't the best environment for getting your money's worth for a costly private tour. The upper floor isn't usually as swamped as the ground floor, so you're in luck if you're interested in that part of the collection.

https://www.walks.com/

My trips to the British Museum have fallen in the months between May and September, so I fear your June experience will be like mine. I will share what a museum staffer told me, in case you have some useful scheduling flexibility: The museum is most crowded on weekends, holidays, Fridays and rainy days. Alas, you're unlikely to have a good precipitation forecast far enough ahead to assist in scheduling a private tour.

Personally, I can't imagine spending 1/2 a day on a tour of the British Museum. It's just too exhausting.

Last spring I took the Around the World in 90 Minutes tour at the British Museum. Our wonderful guide efficiently maneuvered our group of 20 through the crowds to see the highlights. We used headsets so we could hear his excellent commentary over the noise. I highly recommend this tour.