A source of the tubing is your local hobby shop. They stock silicon tubing used for fuel line on model airplanes, boats, and cars.

o_w, I've never grafted tomatoes, I don't live in Britain, etc., etc. But I'm curious and I figured I'd take a look.

For reference, here's what Johnny's Selected Seeds sells (they are a large seed company in the US -- very good people). Apparently the measurement they go by is the inner diameter. Here's their 2 mm:
http://www.johnnyseeds.com/p-6972-silicone-grafting-clip.aspx
and 1.5 mm:
http://www.johnnyseeds.com/p-6971-silicone-grafting-clip.aspx
Johnny's points out that these clips come from the Netherlands, so you'd think they'd be for sale from someone in Britain.

Johnny's also sells spring-loaded clips which are apparently used for varying sizes of stems -- it looks like different growers graft at very different sizes!
http://www.johnnyseeds.com/p-6413-spring-loaded-grafting-clip.aspx
http://www.johnnyseeds.com/p-6414-spring-loaded-grafting-clip.aspx

Here's a British vendor who sells the spring type, which look like the orange clips Johnny's sells for 1/8 - 1/4" stems (the third link above). Oddly, this firm doesn't give any measurements. *frowns disapprovingly* I suppose you could phone them and ask.
http://www.molesseeds.co.uk/flower_and_vegetable_seed_store_uk/Products_Grafting_Clip_4277.html#aVTO650_20_2002

In the course of this adventure, I learned it's possible to use ".uk" in the "Search within a site or domain" field in Google's advanced search. So you might want to try that. Or look at this search:
https://www.google.com/search?q=tomato+grafting+clip+OR+tube+site%3A.uk&hl=en&biw=1372&bih=850&num=10&lr=&ft=i&cr=&safe=images

Good luck and happy tomatoes!

^ thanks for the help. I think I might just go ahead and use plain silicon tubing. I can't see what the point of the bit on the side of those silicon grafting things is though. I wonder if silicon tubing would work as well?

o_w, one of the pages I saw ... somewhere ... showed photos of the silicon grafting clips, including one of a clip falling off the stem when it grew to a certain diameter. Apparently they are designed to do that.

Maybe "the bit on the side" is the mechanism whereby the clip knows when to fall off?

Should have tried a Google Image search yesterday. All sorts of interesting photos....

Here are photos which show pinching one side of the clip to open the "tube" part wider so you can slip it over the stem. That might be the reason for your "bit on the side."
http://vegetablegardendjp.blogspot.com/2010/05/tomato-grafting-with-pictures.html

That blog refers to a video at Johnny's. Johnny's has lots of videos and other helpful stuff. The third video here is about grafting tomatoes:
http://www.johnnyseeds.com/t-video.aspx#instruct
However, that's a different type of clip and a different technique than I've seen elsewhere.

Here's the photo of the grafting clip falling off; detailed info on grafting tomatoes too:
http://en.wikibooks.org/wiki/Horticulture/Tomato
You might also want to read the longer version:
http://www4.ncsu.edu/~clrivard/TubeGraftingTechnique.pdf
[Pardon me while I cheer a bit for my state's university which published that.]

Here's the Google Image search; maybe you can locate a vendor this way. In any case, there are lots of photos and you might find other good info.
Google Image search tomato grafting clip

Good luck with your grafting!

The bit on the side of the smaller clips serves as a handle and you can pinch that side to slightly open the hole on the other side where the scion and rootstock stems go. If you're grafting smaller plants, the silicon clips make it fairly easy to create a joint that won't move as it heals.

If you're trying to do larger plants and you're a patient person, you can use a length of silicon tubing, split along one side so that it will fall off when the stems get large. In my experience, this takes a bit more patience to use (no handles)and I've found that a bit of porous surgical skin tape along one side of the joint can help hold things in position until you get the tubing in place. I'd consider buying tubing in slightly different diameters in case your plants are different sizes at the time you're ready to graft. Ideally, the inside diameter of the tubing should be small enough to hold the joined stems snuggly without pinching.