IANAL either, but it certainly all seems very strange.
My inexpert first guess would be that they're trying to suggest this is a subpoena, which of course it isn't, so they came up with the imaginative concept of "an interview under oath". As to why they think a British citizen who is not a party in any legal proceedings, let alone legal proceedings in Britain, should feel the need to testify under oath at the not-so-polite demand of some American lawyer is anybody's guess. (A demand, it transpires, they sent to Buckingham Palace through Federal Express, who returned it to them as undelivered. Do they genuinely assume all members of the royal family live there, in a sort of Southfork Ranch setup?)
Added strangeness: (1) They continue to refer to their client, Virginia Roberts, as "Jane Doe #3", yet then helpfully include a picture of her in the letter, and refer to press reporting, in which she is of course mentioned by name. (2) If I understand the current state of things correctly, not just Andrew, but also Ms. Roberts isn't as of yet a party in any legal proceedings, anywhere. Her lawyers are merely trying to get an American court to add her as a party in an earlier, ongoing suit against Jeffrey Epstein.