There are lots of competitions out there, some with a really wide choice of categories to enter and others very specific to year groups or subjects. These first ones are national or international competitions; scroll down for ones run by universities and summer schools. Check out the Awards page for even more ideas!
https://iymc.info/en/ (International Youth Math Challenge)
https://isaacphysics.org/ (problem solving competition)
https://oxsci.org/schools/ (essay competition)
www.medicineadmissions.co.uk (for aspiring medics)
www.rgs.org/schools/competitions/young-geographer-of-the-year/ (Royal Geographical Society competition; fixed number of entries per school so you'll need your teachers to get involved)
https://www.bebras.uk/ (computing challenge; held in November)
https://oucc.uk/ (Oxford University Computing Challenge; Feb-Mar - open to top scorers from the Bebras competition above)
http://www.girlsthreeminutescience.org/ (you have 3 mins to get your message across! 1 Jan deadline)
Organisations including National Grid, BP, Shell often run competitions for schools; you'll need to search on their company websites for current opportunities.
There are also Physics, Maths, Chemistry, Biology and Informatics Olympiads which you enter via school.
https://thosethatbreathe.com/ (monthly creative writing competition)
http://ygd.bafta.org/competition (Game Design competition)
bbc.co.uk/nssa (BBC Young Writers’ Award with Cambridge University invites young people in the UK aged between 14 -18 years to submit short stories of up to 1,000 words)
www.bbc.co.uk/youngcomposer (national competition, submissions due in June each year)
www.10xchallenge.org.uk - (start with £10 and make a profitable business in 4 weeks)
www.young-enterprise.org.uk - (run a business for a year under the guidance of a volunteer)
https://www.royalcwsociety.org/the-qcec (essay competition from the Queen's Commonwealth Society)
https://www.esu.org/competitions/psc/ (Debating competition; must apply through your school)
These academic institutions run essay competitions for interested students; check their websites for entry details and entry deadlines, which are typically Jan - April. The university competitions are mainly aimed at 6th form students.
Oxford Colleges and Departments:
https://www.st-hughs.ox.ac.uk/prospective-students/outreach-at-st-hughs/ (History and Classics competitions)
https://www.mod-langs.ox.ac.uk/schools/language-competitions (French, German, Spanish competitions, all age groups)
https://clasoutreach.web.ox.ac.uk/competitions (competitions run by the Classics department at Oxford)
Cambridge Colleges and Departments:
https://www.fitz.cam.ac.uk/admissions/undergraduate-admissions/opportunities-prospective-applicants/essay-competitions (competitions for 6th form students in a range of subjects including architecture, land economy, medieval world, ancient world and classics, history)
https://www.queens.cam.ac.uk/teaching-learning/distinctions-awards/the-estelle-prize-of-ps500-for-year-12-english-students (for year 12 students; closes April; cash prize)
A number of commercial summer schools and organisations run competitions; check their websites for essay requirements and entry deadlines, which are typically March - June.