As Dzug indicates, the US embassy website states that the minimum time to process a visa application, for someone with a criminal record, is '14 to 16 weeks'.
Even if you could get your application processed much quicker, there's an additional problem: The US authorities won't even start to process the paperwork until you provide them with all of the relevant documents. One of these is a copy of your criminal record. To get hold of this, you have to write to your local police HQ, enclosing �10. The police then have up to 40 days in which to respond. So it seems unlikely that you'll even be ready to attend the interview within the next two weeks.
The full, long-winded visa application process is detailed in my post here:
http://www.theanswerbank.co.uk/Travel/Question 289769.html
(That won't help much now but it might be useful for future reference).
There is nothing on the US embassy website about there being any special facility to process urgent applications, such as your own, but it's possible that they might be prepared to consider 'compassionate grounds' for special treatment. I suggest phoning the embassy on 020-7499-9000. However, before you do so, I recommend giving some thought as to how you can obtain documentary proof of your special circumstances (e.g. a faxed letter from the hospital or doctor who is providing treatment for your relative). Without such evidence I doubt that the embassy would be prepared to consider your case.
Chris