How to change the text format of the body field of 200 nodes at once?
The query you suggested is great but if you want to make sure you're only including body fields from the content type you want to update you need to extend it slightly:

UPDATE field_data_body 
SET body_format = REPLACE(body_format, 'plain_text', 'full_html')
WHERE entity_type = 'node' AND bundle = 'flats'


You'll also need to run the same query for the field_revision_body table or nothing will happen:

UPDATE field_revision_body 
SET body_format = REPLACE(body_format, 'plain_text', 'full_html')
WHERE entity_type = 'node' AND bundle = 'flats'