I'm about to travel to Japan (from UK) and can't seem to buy a mains adapter that specifically says it can be used in Japan. However, the internet (e.g. Wikipedia) tells me that Japanese power sockets "appear physically identical" to US ones (although they use a different voltage), so I'm wondering whether I can use an "international adapter" (e.g. this one, with very a useful USB charging feature and multiple socket types) that can be used in the USA, or whether I'll have to wait til I get to the airport and pay more money for it.
You need to distinguish between the physical plug shape and the voltage requirements of the device that you want to plug in.
If you have an adapter that is made for the USA, then yes, in most cases you will be able to plug in that adapter in terms of physical shape into sockets in Japan.
However, you will have to check with each device that you want to use if they support the voltage and mains frequency that you get in Japan out of the power socket. If the label on the device says for example 100-240v, 50/60 Hz (as it says on most laptop power supplies), it WILL work in Japan. If it says 110... or only 240v, then there is high chance that it will NOT work in Japan. Also, you must consider that different parts of Japan use different mains frequencies: 60 Hz which is used in the USA, and 50 Hz which is used in Europe.
The good thing when going from Europe (240v) to Japan (100v) is that you most likely will not ruin your equipment if you plug it in and it cannot work with 100v. The other way round would fry your device in most cases. If you plug it in, it will simply work - or not. If not, you can still try to get an alternative device in Japan for the time you are there. As a very basic rule of thumb: items that have a high ampere requirement (hair driers, toasters etc) will not work. Items that are made to be portable or for travel work in most cases. Check the sticker on the power supply for the "input power" requirements.
What I do when I need confirmation is to check this website:
It has an EXHAUSTIVE list of each and every type, the voltage and the countries that use them. It's been handy for some of the stranger countries I've been to and hasn't failed yet.
Myself, I have a multi-adapter. The only country it doesn't seem to handle is South Africa, which has 3 giant prongs - bigger than the UK ones!
It's worth noting that:
** Although the mains voltage in Japan is the same everywhere, the frequency differs from region to region. Eastern Japan uses predominantly 50 Hz (Tokyo, Kawasaki, Sapporo, Yokohama, Sendai), whereas Western Japan prefers 60 Hz (Osaka, Kyoto, Nagoya, Hiroshima).
In practice, the answer is "yes, usually." If it's OK in the U.S., there's a very good chance it will work without issues in Japan.
Japanese sockets are (mostly) physically compatible with American plugs, and although the nominal line voltage is lower in Japan (100V), the vast majority of modern electronics goods intended for American line voltage (110V) work just fine in Japan. Most DC power supplies have enough latitude that they handle the difference without problems. I've even corresponded with an electronics maker who confirmed this for his product, saying that most power supplies are intentionally designed with a big safety margin to handle brownouts and out-of-spec power anyway, and that the U.S.-Japan difference easily falls within this margin.
The other issue, frequency, is not a problem for most electronics, as they generally convert to DC immediately anyway (it may be an issue for older clocks or anything with a motor driven off of AC).
I've lived in Japan for a long time and used tons of U.S. electronics products here without problems. I think I've only ever had one thing which didn't work properly (an analogue synthesizer with a rare AC power supply that depended on the line voltage for tuning... the digital bits worked fine, but the sound was out of tune).
The item you linked to is a socket adapter, not a voltage converter. It will not convert Japanese standard electricity to UK standard electricity. Therefore, you can only use it with equipment that works over a wide range of voltages. Higher end electronic equipment (e.g. a laptop) and equipment designed for travellers (e.g. a travel hairdryer) might be okay, but you need to check carefully.
By the way, not all USB sockets are created equal. The USB socket of the travel adapter you linked to has a maximum output of 1A. That's not bad but if, say, you charge your iPhone from it, it might take longer than you expect.
You should be right to use the same adapter BUT check the voltage! There's a significant difference between the UK and Japan! See http://www.whichplug.com/uk-to-japan/
I can add to the above. U.K. 240v high current items such as hair dryers and kettles will not work in Japan but it is possible to buy travel types which have a voltage switch 240/120 these should work in the USA and Japan with the right plug adaptor.