The school needs to process the paperwork with the Ministry of Education and get their approval <- you need those documents no matter if you do the ED visa in country at the immigration office OR from the thai consulate in the UK via the eVisa online web portal
You need to decide what you're going to do because IF you're doing the 90 day Non-ED visa from the thai consulate in the UK the documents need to be addressed to that consulate
and
the requirements are different than they are to apply for the 90 day Non-ED visa inside the country.
Read the requirements for the ED visa based on attending a private language school on the website of the thai consulate in London HERE:
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Then you decide if you want to get it thru there BEFORE you wing your way here after you get the documentation from the school
OR
If you wanna come back here and get the 90 day Non-ED visa in country from the thai immigration office that serves the area where you're staying..