You can apply for a job without having had any formal training, especially if you have a good degree and are applying to teach a subject where there is a shortage of teachers. Some independent schools will appoint teachers with no training or experience but plenty of potential, others will not. Some are willing to appoint you on condition that you do on-the-job training (including a PGCE with a university) once you have started.
Applicants from schools overseas are welcomed.
You can find jobs advertised in the Times Educational Supplement.
Be prepared to be flexible. If you are happy to work part-time, say so. If you are happy to work as a teaching assistant for a lower salary, say so. Many such beginner-teachers do the University of Buckingham’s teacher training while doing their job.
If you do not have a university degree you may need to take one.
If you took your degree some time ago or you have a degree in a subject which does not quite align with the subject you hope to teach, consider a Subject Knowledge Enhancement course.
There are a few things you can do to make yourself more employable even without formal teacher training:
If you are keen to work in a prep school (children aged 7-13) then contact IAPS - the Independent Association of Preparatory Schools. They have jobs listed on their website. If the school you are working at is a member of IAPS, the University of Buckingham offers reduction in teacher training course prices.