secret.simon,
The OP may choose to do what he/she thinks is right for him/her. If it can be proved beyond doubt that HO only was at fault in late issuance of CAS thereby making it impossible for the educational institution to issue CAS to OP for him/her to then make an application with CAS included, then the chances of success maybe almost certain. The question is will the caseworker handling the ILR application have the background information to arrive at such a conclusion or to determine who all were at fault or in absence will the caseworker be interested in doing the groundwork to get to the bottom of it or will the caseworker just take the easy route and refuse the application based gaps in application dates that he sees in front of him. IMHO, this case has more to do with fact finding than discretion, so OP has more chances of success in front of a judge than a HO caseworker, provided it was all HO's fault.
Ditima98,
I stand to be corrected but it does appear from
276B(v) that if an out-of-time application made within 28 days of becoming an overstayer is approved then the entire period of such overstay (i.e. - from date of refusal to date of new grant) is ignored for the purpose of settlement under long residency.
Unfortunately for the OP, the application on 13-Dec. was not within the 28 days period and so 276B(v) doesn't help. Hence my comment to the OP regarding "illegal stay not becoming legal".