OK, I just called them again. (Got through on the first try this time-- yippee!) It turns out that I did misunderstand the woman-- she was saying that I couldn't have an appointment before June 2, not that I couldn't BOOK one before then. However, I still can not book an apointment for June 21. Currently they are only accepting appointments through June 5 (2 weeks). I asked the guy if that means they only accept appointments 2 weeks before the date. He said yes, but then he said no, that I should try calling next week, but he was not sure that I would actually be able to book an appointment for the 21st next week. Does anyone know how this works? I seem to remember that the UK embassy in NY had a website with a calendar that showed which dates they were currently booking for appointments, but I see no such system on the Home Office website. It's not a crucial issue, but it would be nice not to waste too much time trying to get through if I can't make the appointment anyway!