British Museum and National Gallery

I don't know that you'll find a semi-private tour offering that combination. The two museums are about 0.8 miles apart (per Google maps), and the British Museum in particular can have a considerable line-up of people at the security check-point. Even if a tour hit the British Museum first, there would still be a lot of time in transit for which you'd be paying the guide but not receiving in-museum guide services. Those museum buildings are large, so just walking from the door to the point where the tour begins and from the end-point back to the door can take many, many minutes.

Have you considered joining separate tours offered by the museums themselves?

British Museum one-hour focused morning tours; these tours begin at 9 AM, an hour before the museum opens to the public. That could be heaven.

Other British Museum tours, at least some of which are free.

Free daily tour at National Gallery, 2PM - 3 PM

In addition, both of the museums have audio guides that provide explanations of some of the objects on display. I don't remember their cost, but it is typically modest.

I imagine you could arrange a private tour of any combination of sights you like, but I think the universe of people qualified to guide you through both an art museum and an ethnographic/history museum is not large.

Though not a combo (many people would not want to spend four to five solid hours at museums during one day), London Walks offers guided tours of both museums: the National Gallery on Fridays and the British museum on Monday, Wednesday, or Saturday.
We took half a dozen tours with this company in 2017 and all were great and relatively small groups except for the Westminster Abbey tour where we had a back up guide for our half of a large group.

LW Nat'l gallery

LW British museum

I agree 100% with both acraven and Nance and will add that I don’t think I could do these back to back. I DO love museums and have been to both a number of times but I need some outside time after either.

2nd vote for London Walks. I’ve not done either of these with LW but did do V&A and thought it was very good. She picked out about 10 items in various galleries to discuss.

I've done both these with different London walks guides in the past. www.walks.com. Their walks last about 2 hours and the guides pick out some highlights. A very good overview.

I’ll add to the London Walks recommendation. For only £10, you get an excellent guide and a small group. They’ll ask at the beginning of the tour what everyone’s interest is and thus they can adjust the tour slightly.

These museums are massive, I’d hate to do both in one day. My preference is always to do two completely different things, one for the am and another for the afternoon. A perfect day for me would be a museum in the am, a neighborhood tour in the afternoon, and theatre or a concert at night, and a couple pints of good beer to separate the periods of the day.

Thank you for the suggestions. I did research Walks, I was looking for some longer tours. I am touring with a company in London and Paris so I am strapped for time. I was trying to squeeze in both on the same day.