Avoid subjects with non-examination assessment (NEA) as they are extremely hard to achieve as a private candidate.
I think you need to pick IGCSE English Language to avoid the speaking and listening element that is hard for private candidates to complete. AQA is more up front about some GCSEs not being feasible for private candidates, the other boards don't ban them but they are very hard to achieve. IGCSEs always have a no-coursework option. Edexcel would be the best bet.
To enter as a private candidate, you need to find an exam centre (=any secondary school or college, plus some private providers) willing to accept you as a private candidate and everything is arranged through them. Expect to pay £100-£150 per GCSE, assuming they are 100% exam and nothing else (e.g. the speaking and listening for English, Spanish oral) and you don't need any access arrangements (extra time, room on your own etc)
You don't have to stick to Edexcel for everything - for instance, AQA is a much more popular board than Edexcel for Spanish and choosing that might find it easier to find a centre to conduct the speaking exam. Edexcel is the most popular for Maths.