If it is a British Council inspected private language school, here are the criteria:http://www.britishcouncil.org/accredita ... idance.htm
It clearly states that the British Council will not comment on your specific qualifications, it is up to your boss to ensure that they meet the description there. Seems you meet the criteria to me.
Here's a description of the minimum requirements for further education colleges:
"TESOL and diplomas will qualify you to teaching in English Language schools and private schools but should you wish to work in further education (also known as the Learning and Skills sector) it gets a little trickier. You will need:
•The Certificate in Further Education Teaching Stage Three at level five (of the revised NQF; Nation Qualifications Framework, more complicated abbreviations!)
•The Certificate for ESOL Subject Specialists at level four.
Qualifications that include these are:
•The ESOL Postgraduate Certificate in Education (PGCE). You will need a degree to do this, it isn’t discipline-specific but a degree relating to languages wouldn’t hurt!
•The CELTA Module Two (Cambridge Certificate in Further Education Teaching Stage Three with the Certificate for ESOL Subject Specialists; what a mouthful!). This is for those who’ve already completed the CELTA course and are already teaching English but planning to move more into teaching in further education."
In other words, no four week Cert is good enough (for a UK further education college) and even someone with a Delta or Trinity DipTESOL will need further qualifications. I think you can take those qualifications while you are doing the job, but note that the ESOL sector is going through huge cuts and so employers are likely to be fussy.