Itinerary Advice

I should also note that flights are booked into London and home from Athens, so those 2 cities are must visits.

Between the two you list, Plan B

If you are arriving in London, I would drop the day in Tuscany and add it to London. If I were taking the trip, as much as I love Florence, I’d split my time between London, Rome and Athens. There is so much to see and do in each of those cities that you would have trouble exploring and enjoying them in less than a week each.

Neither. Your pace seems very unrealistic to me. Unless your goal is a drive-by of all these places. I think you need a Plan C.

It looks like you're planning to fly around a lot. Each time you fly anywhere, you burn most of a day. Keeping that in mind, you only have 1 or 2 places where you're actually in a city for a full day. That's not realistic.

First, start by being honest with yourself: Is Friday June 29th your arrival day from North America? If so, toss that out, you can't count it as a usable day. If that's the case, you have 10 days. And you're planning to hit 6 places (counting "Tuscany" as one place). 6 places in 10 days, eh? Danger, Will Robinson!

With 10 days, I'd pick three places. Yes, three.

You say you're flying in to London, and out of Athens. OK, pick one more place - there's your trip.

I know this isn't what you wanted to hear. Sorry.

I have to agree with David. Neither one of the itineraries are good. Not enough consideration has been given to the travel times between cities. You have almost no time to actually see much of anything before you're leaving again. London definitely needs another day at minimum. And Rome is seriously short changed as well. You need to do some additional editing.

It sounds like you land in London on Friday. Even if you land early in the morning, you're coming off a 7 hour flight, so you won't have had much sleep and you'll probably be jetlagged. By the time you get through the airport and to your hotel, check in and drop your things, it's going to be lunchtime. For most people the rest of the day is mostly a struggle to stay awake long enough to get a decent night's sleep, so you're not likely to get much enjoyment out of the day, other than just experiencing the "vibe." That leaves you one day (hopefully, you will all have had a good night's sleep and be rested) for London. Then you've got to pack, get to the airport with 2-3 hours of lead time (depending mostly on whether you can carry on your bags - budget airports tend to have low cabin bag limits and often strictly enforce them). Then you have to get from the airport (maybe have to wait for luggage?) to your hotel to check in and drop your bags. If you have an early flight, you'll have about 1/2 day to enjoy your new surroundings, get oriented to a new language, new signage, new currency, new transportation system. You get the idea. You don't say how many are in your family. Keep in mind that you will only move as fast as the slowest person, whoever that is at any given moment - whether getting out of bed, out of the shower, out of the dining room, out of the souvenir shop . . . . Lastly, Italy and Athens can be brutally hot with bright sunshine, even in early July, so consider that you may go more slowly than you anticipate.

Which is why you are being advised to limit your stops to 3, or 4 at the very most. Also be careful with flight choices. Not all airports are equal! For instance, London City airport is pretty easy to get to on the tube/light rail, Luton can be a real schlep. Florence has a small airport, it may be hard to find flights. Consider flying into Pisa or Bologna, both of which are reasonably close.

On both plans, I don't understand 1 day in Tuscany. For that one day, what exactly are you thinking of doing in that entire region?

In all of this tight travel, how much of each day are you losing by traveling from city (or region) to city (or region)?

Plan logically and count nights not days. A flight means an entire day for travel, a three hour train in the morning will usually put you near the center of town with most of the day left. Less than three nights stays are a whirlwind where you'll mostly see train stations or airports. LESS IS MORE
Stick to one flight while over there and that'll have to be to Athens because it's really far from London. From London you can take a direct train to Amsterdam OR Paris. Those cities are also connected by a fast train and both are hubs that are sure to have direct flights to Athens. Italy is far and would require another flight so there is another day shot, save it for next time. Based on your flights you'll have three, three night stays and one two nighter.

When you say "Backpacking" what do you mean?

Here's your PLAN C:
Friday, June 29: Arrive - London
Saturday, June 30: London
Sunday, July 1: London
Monday, July 2: London
Tuesday, July 3: Travel Day --> Rome
Wednesday, July 4: Rome
Thursday, July 5: Rome
Friday, July 6: Rome
Saturday, July 7: Travel Day --> Athens
Sunday, July 8: Athens
Monday, July 9: Athens
Tuesday, July 10: Afternoon flight home

I haven't been to Athens, but there is a TON to do in each of London and Rome. With 3 days (4 nights) in each you will barely have time to see the highlights.

I have to say that going to Greece and seeing only Athens is not something I would ever recommend. So my third location would be somewhere else in Greece.

Actually, my next step would be to call the airline and check on what it would cost to change that return flight so your trip is less geograpically spread out. Flying back from London, Edinburgh or Amsterdam might cost very little extra even with the change fees, because Athens isn't usually a cheap gateway.