Recently I had my UK passport stolen and had to fly out to the USA 2 weeks later. I was in Botswana at the time of the theft so I really had pressure.
I was able to apply online for an appointment at the US Consulate General in Cape Town and initially it said the earliest appointment was 4 November. I needed to get it done way sooner than that. I phoned the US Consulate General later that day and was advised that the booking slots had changed and I could get to them on 31 October. First lesson to learn: