Please note the UCAS tariff changed in 2017.
For Logistics / Warfare officers:
'If you take your A-levels after the start of 2017, you need 72 UCAS points, including two non-overlapping subject areas
If you take your GCSEs after the start of 2017, you need 5 at grades 9 – 4, including English Language and Maths."
72 UCAS points is equivalent to DDD at A-level.
What are your target grades in maths/English and your top 3 other subjects?
If you're saying you prefer something a bit more "hands on" then rating is probably your best bet, and would mean you don't need A-levels.
If you're still keen to do the Public Services, then you could join the Royal Naval Reserve (RNR) while you're at college and get an idea whether naval life is for you. However, I'm not sure how applicable Public Services is to other jobs - you may want to choose a course that will give you more options if you later decide not to join the RN.
In any case, go and talk to your local Armed Forces Careers Office (AFCO) who will be able to give you much more detailed advice and help you narrow down what role(s) would fit you best.
Good luck with your GCSEs.