The quickest rail route is via Llanduno Junction (or Chester), Euston & Waterloo. Typical journey times are between 5 hours 50minutes and 6 hours 15 minutes.
The rail route with fewest changes (and avoiding London altogether) is via Manchester Piccadilly and Basingstoke, with a typical journey time of around 7 hours 15 minutes.
Check for details here:
http://www.nationalrail.co.uk/
(NB: To force National Rail's website to offer you the '2 change' route, as well as the 'London' route, make sure that you put a tick in the box marked 'Include slower trains')
National Express offer one coach service per day, changing at London Victoria (where you have 1½ hours to wait). The total journey time is 12 hours 5 minutes:
http://www.nationalexpress.com/coach/index.cfm
Megabus doesn't serve North Wales at all (or, indeed, Andover).
The closest airline route is the Eastern Airways service from Liverpool John Lennon Airport to Southampton but it seems unlikely that it would help you much.
Chris