Transport to Heathrow

Assuming you don't live there and are paying to stay somewhere, I would ask your contacts or hosts or owner of the place that you booked your stay with.

https://www.blackberrycars.com/minicab-to-heathrow/

This sort of service may make sense for your circumstances

EDIT--- inserted general link

Depending exactly on your hotel your closest tube is probably Warwick Avenue or Edgware Road, but you are very close to Paddington, so I wouldn't bother with tube 1 stop from either of those to Paddington. I would just go straight to Paddington (by a very short taxi ride if you want), then Elizabeth line or HEX (to your preference). Perhaps with your bikes HEX is the easiest as totally flat at Paddington, rather than escalators/elevators on Elizabeth.
I have just looked and there are £16.50 advance tickets that day on Heathrow Express- just £4 more than the Elizabeth line, lots of room for the bikes.
But with 6 of you probably a taxi all the way would be the same sort of cost.
Why someone gave you a link for a car from Westminster seems odd, as Little Venice is nowhere near Westminster but go back a step on their website and they serve anywhere in London.

It depends on how much luggage you have and whether or not you can handle all of it plus the two bikes on your own. If taking public transportation it will mean changing trains and more than likely going up and down stairs.

American is at Terminal 3 and depending on where you are going, British Airways is either at Terminal 3 or more likely at Terminal 5.

While I'm a proponent of taking public transportation when you can, in your case, I might just splurge for a car service to take me door to door. A regular taxi will not work as the average taxi can't handle six people, luggage and bikes. You will need a van.

The aforementioned Blackberry cabs is a popular company on this forum. Just airports is another.

With 6 people, luggage and two bikes you aren’t going to fit into any one vehicle so your fare is going to double.
Will your bikes be packed up ready to check on to the plane or will you be doing that at LHR. If the latter I would just take the Elizabeth line or HEX from Paddington. If the former a car would be much easier - maybe bikes + owners in a car and the rest by train?

With 6 people, luggage and two bikes you aren’t going to fit into any one vehicle so your fare is going to double.

Sorry but not true. You can hire a van that can carry more than six people plus full size bags and carry on bags for all. They can make arrangements for the bikes. That's why it is necessary to pre-book.

Will your bikes already be boxed (or are they a folding bike that fits into a suitcase)? You will want to be sure to include the size of the bike boxes and number of pieces of luggage when you pre-order the car service.

I traveled overseas with a bike in a box. I found the bike box to be quite unwieldy so you will want door to door service. Make sure you have any necessary tools. I had to remove the pedals and turn the handlebars perpendicular to fit in the bike box.

In addition to the Blackberry cars mentioned above, there are a couple of other services that have been previously mentioned by others on this board (Just Airports, Simply Airports). I haven’t used any of these specific services, so can’t give a personal recommendation.

How are you getting from the airport to central London? A bike box alone is hellish unless door to door. I speak from experience of many bike trips. Good luck!

Thanks for all the advice. Sounds like a car service makes the most sense. The bikes will be boxed up. My husband and son are the ones doing the bike trip; they are arriving separately from the rest of use (with their bikes). I’m letting them figure out their plans as they don’t seem interested in any advice I pass along from this forum. LOL! They are doing the Lands End to John O’ Groats trip.

Lands End to John O'Groats (aka LEJOG) is a serious bike ride. I assume they are prepared for it. Let them know that the toughest part of the ride is the first bit, getting out of Cornwall and Devon. Some people who do the ride give up by then as they assume it's all going to be like that and expect Scotland to be hillier - it isn't.

V envious. I was going to do it but not alone. My companion bailed. Going in late Aug/early Sept. Not sure where yet!