https://www.passportwaitingtime.co.uk is useful.

If you plan to spend 2 weeks in the UK, bearing in mind public holidays, the 1 week service should be sufficient. This usually takes less than a week (except for children for whom it will take one whole week - seems to be something to do with allowing the other parent more time to object).

If you answer yes to the question about having a passport from another country, you can't use the same day service anyway - unless this has changed in the past year.

As mentioned, as long as your passport is valid on the day of arrival to the UK there would be no problem boarding a direct flight to the UK. Transit may be a different matter although 3 months left may be accepted to visit, not just transit, some countries. You'd have to check individual requirements once you have some candidates for your travel route.