I used Graham Baxter Sporting Tours (now called Baxter Cycling Tours, I think (http://www.baxterscyclingtrips.com/...-de-France-2018.html) for the Tour ages ago.
You'll have to get to England on your own but they arrange everything else
They spent about 5 days in the vicinity of the tour and I rode and watched 3 mountain stages. I did Le Tape de Tour that year (I think they even got my entry for it).

Trek Travel have a couple of options:
Etape Du Tour
TDF
We have done a ride camp with them and were pleased with that.

My wife and I did such a trip as part of our honeymoon in 2012. We used Custom Getaways (a TDF trip operator) and thought they ran a great trip. We rode parts of stages for either 5 or 6 days and then usually watched as the riders went by from some vantage point (a restaurant for lunch on at least one day). They had several TDF-related trips so you had some choice as to the stages you wanted. We picked the trip that had the individual time trial and got to wander around the area where the team vans were and where the athletes were warming up. They also had some mountain stage trips. We also got access to the start line / area and watched them examine Fabian Cancellara's bike *very* carefully. We wound up at the start line where the riders were emerging and were closer to the riders than the TV cameras were.

I did Sport Tours International (UK) last year and had an amazing time. One of my teammates was a tour guide, and I had known others to do this trip. It was their 14 day mountain package, but they have several shorter ones. I thought it would be a bucket list thing, but now I want to go back! I've also known people do Trek Travel. Not the cheapest, but they had a good time. Enjoy!

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"Most of my heroes don't appear on no stamps"
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I'd recommend Ride Holidays - ride holidays.co.nz - a kiwi company that runs awesome tours. High recommended. Even cater for non-cycling partners, if that's applicable

Perfect! Yes, I may have my non-cyclist wife with me. That's good to know.

Thank you everyone for your responses. This is great. Keep them going if there are more experiences and tips out there.

Haute route or cent cols

Book a week with either then go watch the tour

I've used Thomson Bike Tours three times (TDF, Italy and Austria) and will use them again next year (most likely) in Spain. Top notch group. PM me if you'd like more info.

I worked for these guys as a guide in France for about 6 years. They know what they were doing.
http://www.veloechappe.com/

I'll add a plug for https://www.bikestyletours.com/. My wife & I saw the last week of the 2004 Tour with this Aussie bike touring company. Back then, the group leaders were all ex-pro riders. It was great fun riding with Neil Stephens & Alan Pieper. Alan had VIP passes for the last stage & we got to sit in the stands 150 meters from the finish, right behind Robbie McEwen's wife (he won the green jersey that year). There are new owners of the business now, but from what I've read, they still provide great service.