I work in a care home and have just been fired misconduct

Whilst a disciplinary procedure must be conducted without any reasonable delay, failure to do so would only amount to an unfair dismissal. The issue is that if you have been continuously employed at your place of work for less than 2 years then your employment rights will unfortunately be somewhat limited. Most importantly, you will not be protected against unfair dismissal. This means that your employer can dismiss you for more or less any reason, and without following a fair procedure, as long as their decision is not based on discriminatory grounds (i.e. because of gender, race, religion, age, a disability, sexual orientation, etc.) or because you were trying to assert any of your statutory rights (e.g. requesting maternity/paternity leave, etc.).
So you cannot challenge the fact that the employer has taken too long to deal with this matter as that is a procedural matter and you do not have the necessary service to challenge procedural irregularities in the disciplinary process.
I hope this has answered your query. Please take a second to leave a positive rating, or if you need me to clarify anything before you go - please get back to me and I will assist further as best as I can. Thank you