Enter your travel details here:
http://www.nationalrail.co.uk/
The results which initially come up will be on the cross-country route. (Hover your cursor over the number in the 'Chg' column to see where the changes are).
Then to select travel via London, click on 'Journey details' and 'Advanced Search'. Enter 'London' in the 'via' box (selecting the 'All stations' option). Click 'Update train times'.
You'll see that there's probably little difference in journey times, with the London route requiring more changes (and the hassle of getting across London). You'll also see that, in most cases, travel via London is far dearer (but you might happen to come across some bargain 'Advance' fares).
ANY route from Norwich to Llandudno will go via Chester (irrespective of whether you actually have to change there or not), since that's the only practical route!
http://www.nationalra...nalrailnetworkmap.pdf
Chris