We suggest Heathrow Airport as that would give you the opportunity to spend some time in London. From there you could take the train (2 hour trip) from King's Cross railway station directly to Lincoln. To see times and ticket prices, go to https://ojp.nationalrail.co.uk Airports closer to Lincoln would be Humberside, East Midlands, Leeds/Bradford or Birmingham. They are anywhere from one hour to one and a half hours from Lincoln.