Can I either do Medicine or become a paramedic with these grades?

It's possible to do either with those grades, though you'd have to steer clear of universities that put a lot of emphasis on A/A* grades at GCSE, and do well in the other aspects of your application. Your grades are good, but most medical applicants will have better, so if you want to study Medicine you need to concentrate on getting some worthwhile work experience and getting top grades in your A-levels now.

You should also invest some time figuring out whether you'd rather be a doctor or a paramedic! They're pretty different jobs, and unless you're planning on taking a gap year after Y13, you'll need to decide fairly quickly which path you want to go down (so as to make sure you're registered and prepared for the relevant admissions tests, that you've got the right work experience etc). We might be able to help you figure it out here: what is it that appeals to you about those jobs?

I got my GCSE results on Thursday and they're alright, but I already know for a fact I need to carefully consider who I apply with and avoid GCSE heavy universities. I'm also doing Biology, Chemistry, Maths and Psychology at A-Level.

GCSE:
English Literature - B
English Language - B
Maths - B (may be an A, yet to get the remark result)
Further Maths - B
Biology - A
Chemistry - A
Physics - B
World Studies - A*
Drama - A
Geography - B
German - C (getting remarked, long story behind this)
P.E. Studies - Distinction

My teachers all know me at my school as I'm staying on for A-Levels at the Sixth Form, and they're all very confident for me at A-level. But, do I have any chance of going into Medicine or Paramedic Science at university?

For me, what appeals about being a doctor is: the opportunities, the diverse patients I'd be working with, the science etc., working with others and overall it's such an amazing career to go into.

What appeals to me about being a paramedic is: you're not stuck in a hospital, you get to drive around and, again, treat a whole range of patients and you also work closely with others.

If I'm honest, I'm leaning closer to wanting to study medicine and be a doctor. I'm going to work my ass off at A-levels from day one, I've started revising already. Whatever happens I'll go into a career with the NHS, I love what they do. I'm looking everywhere for work experience (I'll be working with a First Responder soon for work experience, applying for bank staff work as soon as I'm old enough (is it 18 you have to be to do bank staff work?), shadowing a porter in a hospital) and I'm currently with St. John Ambulance (Level 3 certified, achieved my Level 1 and 2 in Leadership Skills, going on a Level 1 Peer Education course, going for Cadet of the Year, going on a course for Level 3 First Aid). Planning to get as much work experience as possible to make up for my grades at GCSE.

Unfortunately I can't speak for paramedic science, but I can help with med.
You should be fine so long as you do your research carefully when you're looking for unis. Places like Cardiff have a points system which favour those who got a ridiculous amount of A*s at GCSE, so avoid these. Many others such as Plymouth/BSMS just make sure you meet their minimum requirements and then apply a UKCAT/BMAT cutoff to choose people for interview, so these would be perfect - just make sure you meet the minimum requirements (they'll want either Bs or As in your core subjects at GCSE, and there's a fair bit of variation between unis).
If you can't find anything on their website, shoot admissions an email or call them up to check. All that's needed then would be a decent UKCAT/BMAT/PS and you being on track to nail your A levels, and hopefully you should at least get interviews.

Thank you for your advice, is there any chance you could recommend me a few universities which aren't GCSE heavy, in addition to the two you stated above? Thank you for your help and giving me advice, it is much appreciated.

Thank you for your advice, is there any chance you could recommend me a few universities which aren't GCSE heavy, in addition to the two you stated above? Thank you for your help and giving me advice, it is much appreciated.

I got my GCSE results on Thursday and they're alright, but I already know for a fact I need to carefully consider who I apply with and avoid GCSE heavy universities. I'm also doing Biology, Chemistry, Maths and Psychology at A-Level.

GCSE:
English Literature - B
English Language - B
Maths - B (may be an A, yet to get the remark result)
Further Maths - B
Biology - A
Chemistry - A
Physics - B
World Studies - A*
Drama - A
Geography - B
German - C (getting remarked, long story behind this)
P.E. Studies - Distinction

My teachers all know me at my school as I'm staying on for A-Levels at the Sixth Form, and they're all very confident for me at A-level. But, do I have any chance of going into Medicine or Paramedic Science at university?

They are way above expectations for paramedic science. My friends got C's in GCSE, got DMM in btec applied science and doing bsc paramedic science,

make sure when you do A levels, you get ABB, they will accept you with BBB and a BCC for clearing.

All unis for paramedic science are always in clearing ( meaning there are spots left) every year so i wouldnt worry too much but because many other unis are picking up on it such as st georges ( university of hertfordshrie was the first uni to do paramedic science), its getting relatively popular now.

the reason why its in clearing is because many people tend to not get their driving license before the commencing of the course. I'd recommend you get your license ASAP.

no, you can get a C1 license during/or after the course.

However they do want you to get a provisional c1 license before the commencing of the course. Just for futures sake, if you do paramedic science, You will need to get a D2 form and a D4 form from gov website. They will deliver you the form within a week or two or you can print it online and send it.

You fill the D2 form by yourself while The D4 medical form will need to be filled out by the GP and it will cost ( I paid 60 quid). Send your license, with D2 and D4 medical form to DVLA and within a week or two, you will get your license back and you can check online that you have your provisional C1 entitlement.

Although the requirements on uni websites say you need to get a full manual license + provisional C1 , most unis will be less harsh about the procedure and will allow you to get your provisional C1 entitlement during the course.

EDIT:
I think I deserve a rep