I've been to London a few times and the lineup for the London Eye is always long. I never bothered to see how quick it went though.
A good way to see the city for the first time is to hop on one of the double decker buses. If the weather is good, then you get a great view from the ones with the open top level. You can also get on and off them allday. Pretty much everything is in walking distance, or a short hop on the Underground.
The Strand offers excellent shopping.
Some of the local tour companies offer day trips to places like Stonehenge and Windsor Palace.
Devon and Cornwall offer great country side viewing, but might be a little farther afield than you're looking for.
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