>>My niece lived in France and she suggested Annecy as a visit for a few nights -<<

12 nights only gives you a TOTAL of <u>9.5 days on the ground</u>. Where do you hope to conjure up a 'few nights' in Annecy

>> At this juncture I am looking for a day trip or two outside of London and then the same in Paris it sounds like<<

Then you will have VERY little time in either city.

>>Any additional thoughts?<<

Yep -- listen to your travel agent . . .

Wow, some unnecessarily harsh remarks. Sorry about those.

The way I see it you have 11 nights on the ground.

July 31: fly to Geneva and take the airport bus from GVA to Annecy on Aug 1
Aug 1-2: 2 nights Annecy
Aug 3: direct train or flight to Paris (-or- train to Lyon for a night)
Aug 3-7: assuming going straight to Paris, 5 nights in Paris (can do a day trip to Chartres, Giverny, Rouen, Versailles or wherever interests you.)
Aug 8: Eurostar to London
Aug 8-11: 4 nights in London
Aug 12: fly home

>>The way I see it you have 11 nights on the ground.<<

Possibly, but the OP doesn't even know for sure -- has said several times they have '10 to 12 days?'