re letter-writing, it's a good idea to also include a link to the newspaper article for reference (something I forgot on most occasions)

in the case of Justine Greening, " If you are a Putney, Roehampton or Southfields resident then it would be very helpful if you could include your full name and address in your email. ...

If you are not a constituent but are writing to me in my capacity as Secretary of Education, please visit https://www.gov.uk/contact-dfe and fill out the online contact form, or call the department on 0370 000 2288."

Home Office will reply in ~20 days says the automated response, so it seems the best strategy is to urge Robert Jenrick to make strong representation to the Home Office on behalf of the affected students.

Brandon Lewis has an automated reply which dates from before the election "Parliament has now been dissolved until the General Election. Therefore there are currently no Members of Parliament. Incoming emails to this account may be received and read, however this email address is only being used to respond to urgent constituency case work" But maybe the staff would take notice if there are a lot of emails with similar headings such as "visa extensions for students at Newark College of Violin Making and Repair"