These entries can be added individually using the Joomla Admin->Languages: Overrides->New function but I'm not convinced this always works (probably when "" are not escaped).

Joomla automatically escapes doublequotes in the values when overrides are created this way.
It works perfectly, The doublequotes should never be entered as escaped in the value.

will give
COM_VIRTUEMART_REG_COMPLETE="<div class=\"componentheading\" />Registration complete, these details will be used at checkout. Please continue shopping."

When you edit directly the file xx-XX.override.ini, the escape should be present
Your example IS wrong

PLG_SYSTEM_VPFRAMEWORK_COOKIE_CONSENT_MESSAGE="This website uses cookies to ensure you get the best experience on our website. Please read our <a href="https://www.trendco.co.uk/terms-and-conditions-of-use\" target="_blank">website Terms & Conditions of Use and <a href="https://www.trendco.co.uk/privacy-policy-security#cookies\" target="_blank">Privacy Policy for more information."

the escape sign is only entered for 2 closing doublequotes.

It should be
PLG_SYSTEM_VPFRAMEWORK_COOKIE_CONSENT_MESSAGE="This website uses cookies to ensure you get the best experience on our website. Please read our <a href=\"https://www.trendco.co.uk/terms-and-conditions-of-use\" target=\"_blank\">website Terms & Conditions of Use and <a href=\"https://www.trendco.co.uk/privacy-policy-security#cookies\" target=\"_blank\">Privacy Policy for more information."

Then, it is easy to check that override displays correctly in the override manager (without the escape signs) and no problem at all...

No bug here at all.

I have just edited the override value MOD_LOGIN_VALUE_USERNAME="Email/Email" in Joomla Admin->Language Overrides, changing it to MOD_LOGIN_VALUE_USERNAME="Username / Email" and this resulted in the overrides becoming corrupted same as Actual Result reported above.

I have attached a backup of en-GB.override taken before this occurred.en-GB.override - 20180919 - backup after problem resolved.txt

I have absolutely no issue using your file and then modifying that value.

Can you attach here the "corrupted" file?

I have just repeated this with same result.

Steps:

1. Edit the file /language/overrides/en-GB.override.ini and delete contents, replace with contents in attached file1.txt. This file excludes the 3 values containing escaped quotes.
2. Check Overrides in Joomla Admin - they look good.
3. In Joomla Admin, add the 3 overrides from file2.txt, removing the escaped quotes by changing the \" with ".
4. 2. Check Overrides in Joomla Admin - they look good.
5. Edit any other override in Joomla Admin.
6. Check Overrides in Joomla Admin - all gone except for those in file3.txt. Where do these come from?

file2.txtfile3.txtfile1.txt

I can't replicate this issue wit nightly 3.9 of today. I followed all steps.
Did you clear the cache in overrides view? Maybe a db issue?
Did you check the filter setting?

Absolutely no problem here. No idea what happens in your case.
You may have to totally replace the file on your host by a new one.

In any case, this is not a Joomla bug. Closing Issue.

fyi

I have been using Clear Cache throughout in trying to resolve this problem and checking the contents of table _overrider accordingly. It is currently empty with all overrides restored as per above steps.

I am familiar with the filter setting, Site / Administrator. I am only havving a problem with Site overrides.

File permissions have been checked and are correct.

I have checked the Joomla core files in /administrator/components/com_languages against the same in a new Joomla 3.8.12 package and they are all identical. There are no admin template overrides.