It seems safe to assume they're speaking English. This most likely is an example of a translation convention. As the first two paragraphs of that TV Tropes page put it:
When a group of people whose native language is not English are
together, away from any English speakers, the audience may nonetheless
hear them speaking perfect English.
This is not a case of Translator Microbes, in which an in-universe
device is performing actual translation: We are meant to assume that
the characters are really speaking their own native tongue, and it is
being translated purely for our benefit (or the benefit of the casting
director who is then free to hire English-speaking actors), like the
dubbing of English dialogue onto foreign-made films.
Most of the time, when a translation convention is employed, the writers don't bother indicating that the characters are actually speaking another language, because the audience most likely doesn't care; the audience just wants to know what they're saying.
Further down the page, Death Note is discussed:
In the Japanese version of Death Note, the scenes at which Interpol
convenes and speaks with a British detective, as well as Near's
dealings with the American SPK and Mello's conversations with the US
Mafia are all presented in Japanese; presumably, they're actually
However, in the live-action movie, Lind L. Tailor, L's decoy, was voiced by an American, while a Japanese translator made a voice-over.
This is how it was done in the manga as well, although only the voice-over is shown.
This is assumed to be in effect for Wammy's House as well, unless we are to believe that all the residents of an orphanage in England
speak Japanese as their default language.
There's probably no real evidence anywhere in the series that the characters are speaking English (I certainly don't remember any in the manga). The audience can assume they're speaking English because they're at an orphanage in England. However, as part of the translation convention, we see them speaking Japanese on screen. It's presumed that the audience doesn't care that much to hear them actually speaking English, so the writers just cut to the chase and show them speaking Japanese.
Note, this is different from the situation where L, Near, et al. speak to characters whose native language is Japanese. There, we have to assume they are speaking Japanese (or everyone is speaking English--unlikely, as characters like Misa and Matsuda who probably don't speak English are seen communicating effortlessly with L). But when the Wammy's House members are speaking among themselves, it seems safe to assume they're actually using English and we see it as Japanese under a translation convention.