As promised earlier, here is a brief summary of the structure, and some competitions, in Schools football.
The over arching organisation is ESFA:
http://www.esfa.co.uk/
ESFA run a knock out cup at various age groups for schools and colleges. The rounds start of local, then end up regional, before becoming national. Early rounds will probably/may be officiated by a teacher, however as the rounds progress it becomes more likely that neutral officials will be appointed (although the home side is responsible for appointing the official.). Cards are submitted using a form that can be found somewhere on the website to ESFA and fines will be issued - through the school - to the pupil. ESFA also run some regional leagues, details of which can be found on full time (e.g. here:
http://full-time.thefa.com/Index.do?league=277219506 )
ISFA
http://www.isfa.org.uk/boys/41-boys/regional-tournaments is the Independent Schools FA It also runs a National knockout competition at U15 and U18 level. ISFA is more strict about play by dates and officials, although before the semi finals, it is up to the home school to source neutral officials.
Many counties will then run a School's County Cup. Here is the website for the Somerset County Cup.
http://www.somersetcsfa.org.uk/index.php?page=competitions-2 Most of these games will be ref-ed by the home school. These are for any Schools in the county.
Most counties will run their own County team at U16 level, and there is a national, inter-county knock out Cup for these teams. Neutral refs will probably be appointed, the standard of these teams should be high. Then within a county there will probably be District Schools FA - e.g in Somerset we have (amongst others) Taunton District Schools FA, Yeovil District Schools FA etc., typically at U15s, and (maybe) U16 level, gathering together the best players - often not affiliated to academies - from the schools in those regions. I think these District Schools FA sides are great - run to give boys a good standard of football, for free. The standard is high, the ethic is good - if you can get yourself involved in ref-ing these sides I think you'll enjoy it.