Advise needed- taxi from T5 to Kew and transportation from Kew to Paddington during tube strike

First - nobody yet really knows what the strike will look like, and the different days will be somewhat different.

I have no idea how or where to get a taxi at Heathrow - I've never tried so somebody else will know.

As far as London Transport, use the TfL journey planner website and they will show you a 3 bus or fewer bus only route to Richmond. That will take about an hour but should be reliable.

Can you share where in the Kew Gardens / Richmond area you will need to be?

If I was going from that area to Bath and was unable to get to Paddington I would take the Overground to Richmond then the direct South Western Railway train to Reading at 6 and 36 minutes past each hour- 65 minute journey time.

On our trip to England in May, we stayed in Richmond for 5 days. We arrived at T2 at Heathrow and had no trouble finding the signs for the taxi lineup. Each terminal has a taxi rank. Upon arrival at the line, there was a person coordinating taxis for each party. Our taxi to Richmond cost about £125 so not inexpensive but worth it for us. We waited in line about 10-15 minutes. The Heathrow website offers information and terminal maps to guide you. Hopefully the below link works for T5. Of course there was no strike when we were there so there may be longer waits.

https://maps.heathrow.com/index.html?lang=en&s=eyJvbmxpbmUvaGVhZGVyT25saW5lIjp7InNlYXJjaCI6IlRheGkgUmFuayIsImlzU2VhcmNoQ29uZmlybWVkIjp0cnVlfSwibWFwUmVuZGVyZXIiOnsidnAiOnsibGF0Ijo1MS40NzI1MDQsImxuZyI6LTAuNDg4Njc5LCJ6b29tIjoxNi4yNDkwMDcsImJlYXJpbmciOjAsInBpdGNoIjowfSwib3JkIjo1fX0%3D

With so much uncertainty about the strikes, you could take a wait-and-watch approach, or you could book a car service ahead of time.

I have seen taxi ranks at Heathrow airport, so you could definitely just hail a taxi at the rank outside your terminal. It can take quite a while to exit the airport (~1 hour), so I would not call a cab/Uber while you are deplaning. Here is the link to Heathrow's ground transportation options:
https://www.heathrow.com/transport-and-directions

What hotel are you staying at in Richmond? They should be able to guide you for transportation to Paddington station the next day.

Have a lovely trip to the UK! Don't worry too much about the strikes - it all works out in the end - but give yourselves plenty of time to get around.

From Heathrow what you want is a pre-booked car service rather than a taxi from the rank. Companies like Blackberry Cars and Parker Cars (there are other recommended companies as well) will provide a car with a driver who will meet you in the terminal and escort you and your bags to the car. All for an agreed price a bit cheaper than a black cab and guaranteed no matter what the traffic. they will monitor your flights so know when to turn up.

For the trip to Paddington, whilst the tube is likely to be on strike the regular trains won't be. A train from Richmond to Waterloo will take around 25 minutes. If there are no tubes available on the day just get a regular cab from Waterloo to Paddington. There may also be a clever way of using the London Overground from Richmond as well (they also won't be on strike) but I'm not familiar enough with it to know where to get off. Others maybe.

The clever way by Overground, which is too clever, is Overground to Acton Central, then a 10 to 15 minute walk, then Elizabeth Line from Acton Main Line to Paddington.
But I can't imagine anyone doing that in reality, when there is the direct Reading train from Richmond, which is a lot easier and probably not much slower.
A Londoner would be likely to take the 110 or 190 bus to Hammersmith then the 27 to Paddington.

A schoolboy error from me, not looking at why the OP was going to Paddington. The obvious plan is to take a train to Reading and pick up a train to Bath from there. The only issue I can see with that plan is if you have already have your tickets to Bath and they are Advance tickets. In which case you officially can't pick the train up at Reading, you need to board at Paddington. Now whether, given the planned strikes, anyone would insist on this I'm not sure.

You have great advice from Nigel on your previous thread:
https://community.ricksteves.com/travel-forum/england/preliminary-planning-bath-and-london-before-heading-to-france

Quoting Nigel; lifting one paragraph of his post here:
"Have you considered rearranging the itinerary a little? Getting to Bath on your arrival day is quite easy. If you do Bath first then to the busier London for all the London you can easily take a day trip to the magnificent Kew Garden. Train from Waterloo to Richmond and bus, or District Line Richmond Branch from central London to Kew Gardens station (not step free) and walk to the Gardens through the village, or to Richmond and the number 65 mentioned above."

There is a coach from Heathrow to Bath. You could spend the time you want in Bath; then take the train from there to London. Check into your London hotel. Do Kew and Richmond as a day trip.
This is the easy way to see the places you want.