Don't take this as gospel, others may have differing opinions or more insight, but I'll have a go.
Should we try the buses, or are they going to be so crowded that we can't get on?
Yes, try buses. There will be extra services (hopefully) on all of the central London routes. They will be busy. If you can't get on the first bus that comes, wait for the next one. Don't be surprised by a completely full bus not even opening the front doors. Standing room downstairs, no standing on the stairs or upper deck. If you get on a bus and want a seat upstairs, hover around the stairs, wait for people coming down for their stop, and get up the stairs like a rat up a drainpipe as soon as someone comes down.
Use Citymapper or Google Maps to work out which bus you need. Having a "plan B" bus route that may involve a bit of walking either end is useful. In fact, walking if you're able may be quicker in some cases. The destinations you mention aren't particularly walkable from Knightsbridge though. Reworking your itinerary on strike days to include walkable destinations from Knightsbridge may be an idea if you can.
Everyone else is in the same boat as you, so you just need to muck in and get on with it if you want to go places. It's what everyone that has to be in central London will be doing. Many will work from home or take leave from work on strike days though, but you don't have that option as a visitor.
Try not to travel in rush hour if you can help it. Transport tends to be less busy after 9:30 or 10am. Evening rush starts around 4:30pm and things tend to quiet down after 7pm.
Should we plan to take a taxi, or will they be in such short supply we may not get one?
Taxis will be in high demand, yes. Hard to predict if you'll get one. Every licenced hackney cab will be out on the streets. Don't be surprised if they refuse short journeys. Or long journeys. Or whatever journey they don't like the sound of. I expect that many will be trying the "card machine isn't working" ruse to encourage you to pay in cash for the purposes of tax evasion. Having a working card machine is in the terms of their licence if they're for hire. The drivers will be out to make as much as they can.
Is Uber or Lyft a viable alternative to taxis?
Yes, Uber anyway. I don't think Lyft has much, if any, presence in London. I've never used one. Expect surge pricing on Uber pushing costs up.
And if the strike "ends" at noon, is it safe to plan to take the Tube in the late afternoon or evening?
Expect disruption to tube services for quite a while after the strike ends. Definitely all afternoon. Drivers and trains end up in the wrong place because of strike action and it takes a while to get back on track.
All that said, my intuition is that the strikes will be called off between now and then. The union and its members don't want to be out on strike if they can help it. Dialogue is generally better than it was under the last government, but I'm not party to the nuts and bolts of negotiations so it's hard to call for definite.
Industrial action, or the threat of it, is what the railway unions have to make sure they continue to have good pay and conditions. It's rare in the modern world that a group of workers have that tool and should be supported, even if it's through gritted teeth as you elbow your way on to a packed bus, imho anyway.
Fingers crossed it doesn't happen. Good luck if it does!