First time to London

You do realize that museums in London are free (except for special exhibits from time to time)?
You don't need the pass you mentioned to see those.

No, I personally would not buy a pass.
What sights are you most interested in seeing?
Please list them, to figure out if you would be saving money with any such pass.

You can go to the website for Westminster Abbey, for example, and just buy your tickets for admission.
Likewise for the Tower of London.

Many people find that any kind of pass is expensive and to get to all the sights it includes, you'd have to race around London like a madman.
I found that the pass would have been extremely expensive and not worth it for the sights I wanted to see.

We have a few people on this forum who have used this pass and made it pay off for them.
We also have people who come here from time to time, to advertise the pass.

The only way to know if it would pay off for you is to make a list of your must-see sights and add up what you would pay with regular admission prices.
Many times, after much work, you find it would just be "a wash" or breaking even.
I'm not a big fan of any pass.

Using the pass, to me, takes the spontaneity out of my days in London, as I like to just go at my own pace and see mainly free sights (parks, museums, take walks).

You can buy an Oyster card when you get to London. This lets you "tap in" and ride the tube (underground), city buses, and boats on the river. You don't need for anyone to sell you an Oyster before you get to London.

I have never bothered with the pass on multiple trips to London because realistically, there is no time to hit all the sights in 1 day to make it worth it. I would outline all the places you want to visit that requires an admission fee, add them up and do the math to see if it saves you money. Compare with paying for a 2-3 day pass as well. The museums are free and with your kids ages they may want to just wander around places like Camden Market or Portobello Road Market or Hampstead Heath. In his London guide, Rick recommends several bus lines you can take for a cheap tour of London instead of paying for the expensive Hop on Hop off bus tickets.

Do you recommend a black cab tour?

No. This would be very expensive. And you would basically see the cars that are stopped alongside your cab in the thick London traffic.

There are Hop-On-Hop-Off bus tours I would consider instead. You're sitting up high--get seats on the top floor of the (open top) bus.
Many people say not to do this tour, you'll get stuck in traffic, but perhaps it suits what you're looking for.
There are several bus tour companies; all listed in the Rick Steves London guidebook.

I highly recommend getting either of these guidebooks. The information in them will pay off during your trip.
Pocket Guidebook:
https://store.ricksteves.com/shop/p/pocket-london
Full size London guidebook:
https://store.ricksteves.com/shop/p/london-guidebook

You can see a lot of London by taking city buses, instead of the Hop-On tour.
Pick up a bus map at one of the tube stations and consider riding these.
Just tap in with your Oyster card, which you get from a machine at the tube stations.

The Rick Steves book is definitely recommended, and although I have been to London MANY times, I did an audio podcast tour of his and learned all sorts of things.

Great intel by Rebecca. Take it to heart.

Definite NO on London Pass or Black Cab tour.

Get a copy of the RS London guide. Lots of pertinent info to help plan a comfortable itinerary.

Continue to ask questions on this Forum as well as using its Search engine. Click on the 5 horizontal on the top left of the header. Search Box will appear.

Make note of the following:

Cross streets at lights. Don’t jay walk. Traffic flows the opposite to what your brain is used to. Can’t tell how many times I’ve seen near misses when tourists attempt to jaywalk.

Majority of museums are free ( but if you can, toss a pound into the donation box)

Pubs serve good food.

See theatre.

Stroll in parks.

Great city to enjoy with your sons.