I got no offers for university for engineering what should I do!

Im wanting to pursue aerospace/mechanical engineering at University of Sheffield, my home city, im not eager to move out since im not confident with that idea! Though I got no offers applying for a foundation year for 4 different engineering courses, I am predicted ABB in biology chemistry and physics though sure I could achieve A*AB/A if I put more work in. I only achieved a grade 6 in GCSE math, my gcses were pretty mediocre.
What would be best to do? Pursue something else, take a gap year and do A-level math and apply next year?

Ill be blunt, maths is crucial to engineering. Applied physics & engineering is described with maths.
Im guessing (youll have to check) you got automatically rejected based on your GCSE maths score. Typically maths should be taken at A-level to study engineering.
If you want to work in engineering I would suggest:

Thank you for your response,
I was aiming to take a gap year and build my knowledge of mathematics. Would it be necessary to complete gcse math again to achieve a better grade or shall I complete some prior knowledge of it since my foundation is great after most of it being taught within alevel physics, and just do alevel maths?

Sorry to hear you have not received any offers, its very demoralising
Engineering is a huge field, be careful not to assume it is all highly mathematical - you haven't chosen A level Maths so it may be you should be looking at engineering related subjects that play to your strengths.
What are you good at? What do you enjoy?
An example, I was good at Physics and interested in the design side of Engineering. I did Materials, Engineering and Design. Turned out I had a talent for metals and corrosion, ended up doing that for a while
Another key question is about you wanting to stay in Sheffield - that limits you to 2 Unis (yet you applied for 4 Foundations Degrees?)
A big chunk of the reason to go to Uni is to go away from home, grow yourself and your confidence. Sheffield is within an hour of lots of Unis, so you have plenty of choice
Your other question - what to do next? - IMO look at UCAS Extra (I dont know much about it) and start research for Clearing.
I know some uni admissions people well, and lots of unis are worried about under recruiting this year, especially the mid field unis in financial trouble. I am certain you will find something in Clearing on results day, but you will need to be prepared (ie know the unis and courses you want and have their phone numbers with you) FWIW many unis start putting courses in Clearing on their website well before Results Day

I had applied to: chemical, materials, aerospace, mech eng and also pharmacy as I was skeptical to apply to engineering and wanted a backup of some sort if I didn’t want to go the extra mile in pursuing engineering. I hope my alevels are good enough to go through clearing and hopefully end up in an engineering course at Sheffield though my gcse grade is below the grade requirements of 7.

Applying for a degree without meeting the minimum GCSE requirements is a principle error, your advisors should have informed you this was not a sensible approach.
Additionally applying for several courses at te same university doesn’t increase your chances of an offer- it was almost certainly the case for the engineering disciplines you get an offer to all or none of them, and actually applying to all them might create uncertainty in your peak interest in the subject you are applying for and reduce your likelihood of an offer.

I felt very rushed with sending my ucas application, my sixth form was supposed to check if thrice if there was any issues with it though I don’t think it was done even on my behalf of checking it. I assumed a missed gcse requirement would be fine as long as my grades in alevel were okay. I’m deciding for a sensible option whether doing a degree apprenticeship would be best or uni, though I’m unsure if I can do gcse math resist and alevel math at the same time.

I felt very rushed with sending my ucas application, my sixth form was supposed to check if thrice if there was any issues with it though I don’t think it was done even on my behalf of checking it. I assumed a missed gcse requirement would be fine as long as my grades in alevel were okay. I’m deciding for a sensible option whether doing a degree apprenticeship would be best or uni, though I’m unsure if I can do gcse math resist and alevel math at the same time.

The AMRC has some good engineering apprenticeships that you could consider.
Uni of sheff goes into clearing for most courses, I'm not sure about engineering though.
I don't think its to big of a deal that your one grade below. I know plenty of people that got in even though they didn't entirely meet the GCSE requirements.
Hope it works out for you.

Im wanting to pursue aerospace/mechanical engineering at University of Sheffield, my home city, im not eager to move out since im not confident with that idea! Though I got no offers applying for a foundation year for 4 different engineering courses, I am predicted ABB in biology chemistry and physics though sure I could achieve A*AB/A if I put more work in. I only achieved a grade 6 in GCSE math, my gcses were pretty mediocre.
What would be best to do? Pursue something else, take a gap year and do A-level math and apply next year?