I'm wondering if and how soon ukba may send out a letter saying they know I've overstayed?
There's no way to know, I'm afraid. They don't keep tabs on everyone on visitor visas to make sure they've left and so I don't think they would send out a letter.
It's your responsibility to know what date you must leave by and to make sure you are no longer in the UK after that date.
If you do overstay, it is then also your responsibility to declare your overstay on any future travel applications to the UK and to be completely honest about it.
- If you leave the UK within 28 days of the visa expiring, then you shouldn't be refused a visa in future (though you will need to apply for one in advance before trying to come back as a visitor, since you now have an overstay on your travel history)
- If you overstay for more than 90 days, you can be deported (which can result in up to a 10-year ban from re-entering the UK)