Will they send me a letter requiring me to leave the country within 30 days? Or will this only affect future visa applications?
Will they send me a letter requiring me to leave the country within 30 days?
The UK cannot know for sure whether or not you've left the country as you could've traveled to Ireland overland without this being recorded anywhere. So they won't be sending out letters, unless you happen to get caught in a random document check, which is quite rare.
Or will this only affect future visa applications?
It will affect both visa applications and future attempts to enter the UK. It can also jeopardize your future travel to countries like Canada, Australia, US, New Zealand, etc, which ask about prior refusals when applying for electronic authorization to travel there.
My general advice is... do not overstay. I understand this might seem like a reasonable course of action right now, but this is a good way to mess up your future travels. Leave before the 6 months period and don't violate immigration laws during your stay. If you're worried about re-entering the UK if you leave right now and immediately try to come back, you can choose to travel to the UK via Ireland which all but guarantees you entry into the country. See my answer on a related question for details.
The general (unofficial) rule is that you should only return to the UK after spending the same amount of time outside of the UK as the time spent on your previous visit (which may be less than the granted leave to enter).
Immigration Rules - Immigration Rules Appendix V: Visitor - Guidance - GOV.UK
Genuine visitor requirement
V 4.2 The applicant must satisfy the decision maker that they are a genuine visitor, which means the applicant:
(b) will not live in the UK for extended periods through frequent or successive visits, or make the UK their main home; and
An existing leave to enter is not effected by a stay in the Republic of Ireland. (CTA rules: Exemptions from deemed leave on the basis of specific status).
It is up to the discretion of the Immigration Officer to grant a leave to enter.
Even if you do not overstay by a few days, any return before 6 months may be denied since it would look as if you are trying to stay for a prelonged period after just spending 6 months in the UK.
Returning after 7-8 months, the Immigration Officer may see the overstay of a few days but may choose to ignore it if they have the impression you are not trying to stay for a prelonged period.
The situation would change if you were found in the UK while overstaying.
That would have a serious effect on any future reentry.