1. Pursuant to your application being prior to the expiry of your student visa it is clear that the regularisation scheme was not applicable since you were not an overstayer. I am therefore unable to ascertain what form of leave you/ your representative applied for - do you have copies of the paperwork. What was asked for and on what grounds? I can only assume you applied for ILR (I'm giving you the benefit of doubt) on a misunderstanding/misrepresentation of the regularisation scheme. This application was refused since:
(a) the regularisation scheme did not apply to you, and worse
(b) it came to the attention of the HO you were in breach of the rules by working full time and that you did not intend to leave the UK at the end of your studies.
2. Under the immigration rules a condition for grant of ILR under the 10 year rule is that you have lawful residence:
276A - HC395 wrote:
(b) "lawful residence" means residence which is continuous residence pursuant to:
(i) existing leave to enter or remain; or
Under Sec 3C of the 1971 Act as amended your leave was extended to 10 working days after the recipt of your decision = the same time you have to submit an appeal if your application is refused which it was. You have no formal leave at present - Sec 3C does not apply
to the period of appeal i.e. you fail to meet 276A (b)(i). Likewise it does not apply to fresh applications post expiry of your leave. Hence your time in the UK since the refusal = Jan 2003 is not considered in the 10 yr period..your clock went back to zero and has stayed there. However you are not an overstayer pending the HO making a decision on the appeal.
3. The appeal is a mere formality because you applied for ILR when your student visa was valid. The HO have reviewed your application and found it to be without merit. On what basis do you expect the appeal to succeed given that the facts of the case hold? The mere fact of breaching your conditions by working full time means your case is a non starter. Your Human Rights must not circumvent immigration controls - the circumstances you describe i.e. kids is not a compelling enough ground because they can accompany you to your home country. Even where the children have settled status in the UK the courts have argued that they can if young enough settle abroad successfully with their parents. You have the rights to be with your kids but not necessarily in the UK is the theme here.
If your appeal is successful then you may apply for ILR as applicable if this is not what is granted since such leave forgives/ covers the period Jan 2003 to decision date. Sorry this is not the news you wanted to hear. Of course its only my opinion - others might think/ believe otherwise.