It is worth reading IPSEA and SOSSEN’s websites so you understand the processes and law.
If this is a first EHCP, the LA must finalise within 20 weeks of the EHCNA request, or if you had to appeal refusal to assess but not issue, within 14 weeks of the conclusion of that appeal, or if you had to appeal refusal to issue, within 11 weeks of the conclusion of that appeal.
If this is following an annual review, the LA must finalise within 8 weeks of informing you they propose to amend/sending you the amendment notice so should be within a maximum of 12 weeks from the annual review meeting.
If the LA is in breach these timescales, you should chase now. IPSEA has model letters you can use. If they aren’t yet in breach but would be if you waited until June, remind the LA of the timescales.
Not relevant if your preferred school hasn’t objected, but just so you know, non-wholly independent schools can still be named even if they object.