The 1.5 mile isnt untill week 5(along with the pressups and situps i believe) so i wouldnt be too worried as you will have plenty of fitness related training before hand, if you cant do it by week 5 then its not your fitness that is lacking, but your will. Even if you fail it then you have up to 2 weeks of remedial training (which goes into your free time, another reason not to fail as theres not much free time to start with) where they try improve your 1.5 mile time. Not everybody who joins up at the same time as you is going to be fit and there will be atleast a few people in the same situation as you. I could have what i said a bit mixed up as i cant remember too well, you do a 1.5 mile in your first week and you may be remedial training if you fail that, either way its achieveable.
I got a time of 9.27 for my 1.5 mile in raleigh which wasnt that bad and i did bugger all during phase 2 and at the end did my mile and a half again and got 8.54.....figure that out.
Overall there isnt really that much to worry about but it wouldnt hurt to start running a bit now, you've left it a bit too late but that extra bit of practice wont do you any harm.
As soon as you get through phase 1 training the rest is a piece of cake. I really enjoyed phase 2 and its so relaxed by comparison. Just keep your head down and try not to get into any trouble and the weeks should fly by.
edit: think it is week 6 like has been mentioned, dartmoor is week 5