Schools can currently choose a route to GCSEs from a number of examination boards and a variety of specifcations under each board. This is not the case, however, for private candidates, for this reason: their 'missing-out' on coursework or internally-assessed assignments, which schools only can organises and supervise. There is little point in a private candidate's registering for an examination under a specification in which, say, only 60% of the assessment is on final exam performance.
The private candidate therefore has a restricted range of possibilities, with the condition of setting everything on performance in two written papers per subject, held a week or so apart. The qualifications in English and Maths I am recommending to my students (not-in-school, or mature) are the Edexcel IGCSE's. Both foundation and higher tiers are provided for and the assessment is 100% by final exam. If they go ahead, Kent College, Canterbury, will be the local examination centre.
Other options for private candiates may be available, but I haven't come across them so far.