ok, writing quite frankly here.
as a sixth form tutor and in charge of admissions for several subjects, this is the situation with cancer.
A certain number of children with cancer will miss time in the sixth form or even drop out.
This severely affects the retention and success criteria of individual teachers, who rely on their statistics for job security, and to be able to pay their mortgage, quite frankly.
There is ABSOLUTLY NO reason why a student cannot start year 12 age 17, many many do, mostly becasue they have messed up and are restarting in different subjects in different schools.
Some because they spent a year retaking GCSEs, or doing a BTEC level 2 course to qualify for a BTEC level 3 course, or sometimes have done well enough on BTEC to transfer to A levels.
However, a child with cancer is going to be something many teachers will strive vigorously to avoid - please don't judge too harshly, they are thinking of keeping a roof over their own children's heads.
You need to contact the chair of governors right away,
but be all nice, as nice as possible, you don't want to be aggressive, or repeat what I have told you, as it will set up a confrontation which will not be constructive.
get onto them right now, and just keep on and on until they let her in, they have no legal reason to refuse.
Once she is in, she is likely to be treated with sympathy and inclusion by staff, who of course care for teenagers, or they wouldn't be doing that job, and of course will care particularly for a child who has had such a hard time.
once she is in, and they have accepted that, that is, they won't want her on their register and on their statistics, but it isn't personal, and once she is, there is unlikly to be any issue
I hope that makes sense.
I have come across similar many times, in fact ended up being asked to offer places to many students with cancer, on the basis that my mortgage was paid and I had no particular career ambitions, which was fine, if they actually wanted to do my course!