In June I will be arriving in London and flying out of Rome with 3 weeks in between.
So far, the places I know I will visit:

London
Berlin
Vienna
Florence
Rome

I am considering going up to Edinburgh for a few days, but will it be worth it given how little time I have?
Not sure how long I should spend in each destination,
and I'm also keen to visit other cities in those countries that aren't the major cities.

Any advice is greatly appreciated!!!!

Rome to Florence is the only really easy connection on your list.

The rest you'll either fly or need longer train rides.

IMHO once you back out travel you should be thinking about dropping a stop not adding one. But it depends on what you intend to do and see in each city.

Twenty one days with five stops sounds easy but you've got Four national capitals. Three used to capitals of empires. You've got to get between the five stops. Are you at least flying open jaw (multi city)?

Yes mainly fly between those bases - take trains around Italy and maybe the overnight train to Vienna - book early for dicounted tickets - www.seat61.com has loads on doing that - general info trains BETS-European Rail Experts and www.ricksteves.com.

Edinburgh - too much an outlier and if wanting to visit other cities in those countries just little time - a fly by at most if other places ae a must.

How I have been planning my holiday is
Arrive - no planned activities
city
city
Travel day.
city
city
What I mean by that is for every day you travel into a new country, I pretty much discount the whole day. Other than Venice to Florence, i'd discount 1/2 a day.

I used visit a city, forums, books, guidebooks, Rick Steves, Youtube, reviews, friends, family everything to plan my trip. I'm an over planner. I'm still in the planning process now (The beginning is very stressful, but once you get it the places decided it gets easier). I even used Viator (even though I won't book with them directly, it allows you to see what trips are available or sights then you can book directly with the company or make your own way there). Visitacity is a good website.

Decide on what you want to see in each city and that will help you decide how many days you will need in each place. For instance, I was initially only planning on going to Krakow for a day or two because I really had a desire to go to Auchwitz and knew I needed to see that concentration camp, no one really speaks about Poland and no one I know has been, so it wasn't until I read online and people suggested staying longer, that I really researched the area and now I have planned so much history such as, Schlinders Factory Visit, Pharmacy under the eagle, Gestapo HQ museum. Once I decided that there was so much more to see, I planned it in my excel document and decided on the number of days. Also note, that peoples interests will vary from yours, my husband wants to see Belgium and wants to have a relaxed trip and despite my best efforts to convince him to drop a day, he won't, even though it looks boring to me, he seems to think it will be interesting.I do hope I am wrong. So bare in mind, someone might suggest 7 days in Rome, others might say 3 days was enough etc. I do know from reading that the Vatican City takes a whole day, so keep that in mind when deciding your length of time.

5 cities (your original wish list) in 3 weeks is plenty, maybe more than enough. No I would not try to add another one making it 6 (spread out) places in only 21 days.

Reiterating MaryAlex' comments... You need to do sightseeing research to decide what you want to do/see in each place and then try to figure out how much time that will take. Since we don't know you, we can't tell you how much time you should spend anywhere. As a first-timer, you could easily spend a week in London and a week in Rome without running out of interesting sights. There are multiple day trip options from all five destinations that you could consider to contrast the big city experience with smaller, quieter breaks.

London and Edinburgh are my two favorite cities from your list/possibilities. But No, I would not try to squeeze in Edinburgh. You have more than enough to fill three weeks and that will be pretty rushed. If your entire trip is 3 weeks/21 days -- then you will have 18.5 days to play with and all travel days will come off that. So about 15 free days for seeing/doing.

But if you have 21 full days on the ground -- (a 24 day all in) it will be a tad easier but still pretty fast paced. And if the 'three weeks' is really 'about 3 weeks' like say 19 days -- then you'd be down to about 12 or 13 days free. So exactly how many days do you have?

Thanks so much everyone, this is really really helpful!

I essentially have 22 days from arrival to departure, and yes I'll be flying open jaw.
And yes I'll be doing as much research as I can so I can be fairly productive with my time when I arrive at each destination.

This is the current breakdown that I'm aiming for IF i include Edinburgh:

London - 5
Edinburgh - 2
VIenna - 5
Berlin - 3
Cinque Terre - 1
Florence - 4
Rome - 2

You aren't counting days correctly, as you aren't counting transportation. You will lose a LOT of time -- sometimes the better part of a day -- each time you change locations. In addition to the actual transportation time, you need to add some time on either side for getting to/from your lodging, checking in/out, packing/unpacking, getting lost/oriented, etc.

And even if you could manage some kind of instantaneous time travel to get from place to place, 3 days is VERY little for Berlin and 2 days is next to nothing for Rome.

Good luck!

Your list is not 22 days - not even close. To get the actual number of days in the various places would require 27 nights on the ground.

So now, where do you want to cut?

To illustrate what KJA and JanisJ are saying, let's consider the Cinque Terre. You will have one day, according to your plan, in the Cinque Terre. The day before you'll be in Berlin and the day after in Florence. With this itinerary, you'll arrive in the Cinque Terre late in the day, and you'll leave the next day for Florence. Depending on which town you'll be staying in, you'll have a three or four hour journey by train, with two changes of train to get to Florence. At most you'll have a few hours in the Cinque Terre, and two days of travel to get there and get away.

You have to begin counting nights in each place, not days, and consider that two nights really means one day, or one and a half days if the two places are near each other.

Listen to what kja, janisj and bvlenci are saying. They know what they are talking about. If you look around through this forum you will literally see a thousand posts saying that and for good reason.

Since you were undecided about Edinburgh I would scrap that. Then I would scrap at least one of either Berlin or Vienna as they are geographically further apart. They are all worthwhile destinations, but you have to think about how long/difficult it is to get from one point to the next. You are starting in London and ending in Rome so those are givens. Even if you just did England and Italy you will need to fly once in the middle of your trip. You will have to add another flight for each of Berlin and Vienna.

In fact given your starting and ending points and time frame, a really nice trip would be about a week in London with some day trips and the rest of the time in Italy. The CT (and area around it), Florence (with day trips) and Rome would be great in the amount of time you have. Even if you are a really 'fast' traveler and get bored easily, there are so many day trip options from those bases that you won't run out of things to do.

How can you do the CT in only "1 day" allowed in between Berlin and Florence?