If, as happens in the majority of cases, the issue is very clear and the LA concedes on receipt of the pre-action letter, that could mean the problem being resolved within two or three weeks.
If you do have to take it to the next stage, solicitors can apply in urgent cases for emergency legal aid – and where a child is not receiving education or SEN provision, the Legal Aid Agency is prepared to treat it as urgent. That means that legal aid could be secured within a week or two of solicitors being instructed. It might be limited to something like requiring counsel’s opinion in the early stages, but again that can normally be dealt with quickly. If or when Legal Aid is granted, notice of that fact has to be served on the LA and sometimes that alone persuades them to concede.
If you have to start proceedings, how long it will take depends on the circumstances and the court’s backlog. In theory a fully defended claim may take several weeks to come to hearing, but this is rare; only a tiny proportion of defended JR cases get that far because they tend to be settled.
However, if the issue is urgent, you may be able to ask for ‘interim relief’: i.e. a temporary order granting what you need – for example tuition at home – until the main action itself is heard. This is normally arranged by way of a shorter hearing which could take place within 2-3 weeks of the action starting, or even sooner in a really urgent case.