Home educated children often choose to follow GCSE courses at home, perhaps by way of distance learning courses offered by establishments like Oxford Home Schooling.
Unfortunately it is a common problem for EHE students, that once they have completed the course and are at the point of exam readiness, they cannot find anywhere to actually sit the exam.

If you are registered or don’t mind contacting your local authority EHE team, they might have a list of schools or colleges in your area prepared to take on external candidates.

Alternatively, you could approach colleges in your area and explain to them the difficulty you are having and ask if they are able to help. Also, independent schools can often be accommodating, especially since now they are encouraged by the government to offer ‘out reach’ programmes to help benefit the wider community.

Adult education centres can also be helpful.

Finding external exam centres for EHE students can be a real problem for home educators, which many home education groups and organisations are trying to tackle. A lot of home educated children take Open University courses instead or just wait until they are 16yrs+ old and enroll at a college (gaining entry through port folio and/or entry exam).

Below is a list of a few places that HESW is aware of that will accept external EHE candidates:

Somerset

Richard Huish college, Taunton have established an exam centre for EHE students. GCSE exams can be taken in any subject, any board.  They advise talking to the college at the start of the course you intend to study, as they may be able to offer advice and assistance for you to achieve the best outcome.  There is only a very minimal administration charge, currently (2010) £2.50, plus invigilation costs (divided amongst the number of students present), plus of course the actual exam fee set by the board (£25 - £45 depending on board and subject).  Somerset LA may pay the costs of taking exams for EHE children who are registered. 
Richard Huish will accept candidates from outside of Somerset.

3A Tutors Ltd is a private tutorial college and independent examination centre  based in Bristol. They specialise in the provision of external candidate  examination entry and a range of optional tuition services for  home-study and mature students from across the UK and overseas who wish  to achieve recognised qualifications in one or more subjects.

Devon

The Maynard School in Exeter is happy to accept private candidates to sit GCSE, IGCSE and GCE examinations in most subjects, with AQA, Edexcel or OCR. They have November, January, March and summer sittings.  Candidates usually join the school’s students in the examination hall (and are happy to include boys, even though it is a girls’ school). Separate invigilation can be arranged, although there is an additional fee for this. They are also happy to arrange for extra time (with the correct evidence presented in advance). Coursework is not marked, but they are happy to forward it to the Awarding Body for marking. The school can offer ISA test for AQA GCSE Sciences, and oral tests in some Modern Languages (if they are with the same Awarding Body as their own candidates).
As well as the exam board fee (£25 - £45 depending on board and subject), there is an additional administration fee which is about £30, with extra for oral tests etc.  A small supplement is then added for extra subjects in the same exam session.
There are no restrictions on the age of students.

Bicton College - are open to taking home educated students and have some good vocational courses for 14-16 year olds: Agriculture, Animal Care, Land-based Countryside, Practical Gardening, Sport and Recreation, Horse Care, Construction, Motor Vehicle.  Students can enroll with attendance of 1 day per week following an academic year.