London-Amsterdam book your reservation for Eurostar in advance, they can get soldout soon for Passholder.

Amsterdam-Berlin there are direct IC trains, reservation optional. No Nighttrains, you could use a ICE/IC train to Duisburg or Düsseldorf and from there an ICE train that goes by night to Berlin, only seats, reservation optional.

Berlin-Zermatt: Nightjet Berlin-Basel (reservation compulsory for couchette and sleeper, seats run as IC with optional reservation) and from Basel IC6 trains to Visp, change there for regional trains to Zermatt. If you have time you could go from Bern with the RE via Lötschberg scenic railway, or travel via Gotthard line to Andermatt and from there to Brig and Visp.

Zermatt-Venice: Why a nightmare? Timetables for Italy after march are not online, you can plan with timetables of now. There is one direct Eurocity (EC) train every morning from Brig (7:40) to Venice, or you have to change in Milan for later EC trains. If you like to do a scenic ride please leave a comment, but you have to change trains some time.

The Eurocity train Switzerland-Italy needs a reservation in Italy (from Domodossola).

Venice-Lake Bled:

Regional Expres (RV) trains Venice to Gorizia Centrale, there change trainstation from Gorizia Centrale in Italy to Nova Gorica station in Slovenia (walk, bus or taxi). From there local trains to Bled Jezero station. If you want to go to the other side of the lake you have to travel further to Jesenice and change there to the first train to Lesce-Bled or bus/taxi/walk.

You can also travel via Trieste Centrale (scenic railway near the adriatic) and Ljubljana. Change in Trieste and Ljubljana.

From Lesce Bled there are direct Eurocity trains to Zagreb. And from there trains to Split. Trains to Split have compulsory reservations, that can't be booked from outside Croatia. So keep in mind to have time to book at a station. All other trains are only optional or no reservation needed. There is also a nighttrain Zagreb-Split.

There are 3 routes through the alps to Italy - The 2 obvious ones are Munich to Verona (and beyond) and Vienna to Venice. However to get from Venice to Split may involve a challenging routing.

https://www.seat61.com/trains-and-routes/venice-and-trieste-to-lake-bled-by-train.htm

Bled to Split appears to be via Zagreb and connections can be quite long. Others can advise more on that. My memory of Split (on a cruise) is that one day would be more than enough, so unless you have a good reason to stay a week…...

Normally I advise against sleepers for several reasons (personal preference and cost). With 3 the cabins would be cramped or you have to sleep in couchettes (still cramped but little or no privacy). There are few sleepers equipped with en-suite so you have to parade to the end of the carriage. For relative comfort the cost for 3 can be more than a reasonable hotel/apartment.

Far better to stay in family accomodation (hotel or apartment) and travel in daylight to enjoy the journey. For Venice always stop outside the main Island (I usually stay in Mestre - 10 mins by train or 20 mins by tram) for prices much more reasonable.

You might consider a stay in Hamburg, not least to see the model railway.

We found for simple city visits an approach of travel day 1, arrive pm, evening in city, hotel (2 nights), full day in city then day 3 morning journey to next stop, so you may need to limit your expectations on number of destinations. For one stop we stayed 3 nights in Nuremberg and toured Bavaria on 2 days (plenty of interesting places in easy train reach inc Bamberg, Amberg, Regensburg and even Munich.) There are direct trains from Berlin to Bavaria.

We came up from Sicily (via Salerno and Bologna) then through the Brenner pass direct to Munich. 

I would probably avoid Amsterdam unless it is your first visit there - it really needs a dedicated visit, and is a bit of a dog-leg for Berlin.. If you go through Brussels you can connect with the German ICE to Cologne and Frankfurt for onward to Northern Germany.

Good luck with your planning 

Some extra tip. Don't visit Bled, but go for the lake of Bohinj a couple of kilometers further in Slovenia. Get off in Bohinjska Bistrica. It's the same train as @Angelo suggested. Much less instagram tourism, more natural and plenty of outdoor activities to do which your children may like very much. In summer there are quite some free busses to the lake and into the mountains.

You're planning a bit in advance, no panic if certain connections do not show up. Some are only available a couple of months before departure, same counts for buying reservations. Usually 3 months before departure most should be available.

Best use planners like the one of Deutsche Bahn or ÖBB. They cover a lot of Europe. 

Good luck planning! 

For Your Route from Berlin to Zermatt you may catch the Nighttrain to Zürich till Basel or all the way to Zürich (longer sleep) 
Then you can choice between the fast way via Visp to Zermatt. or a scenic Route via the Lötschberg Hill Route to Brig and then to Zermatt. 

From Zermatt to Venice is pretty simple just catch a Train to Visp/Brig and then use the direct Eurocity to Venice :) 

and yes i prefer as @BrendanDB  Lake Bohinj over Lake Bled. Another popular thing is the Postoinja Cave Park :) 

Thank you all so much! I’m still getting my head around working out routes etc but this is really useful with some great tips - really appreciate the help and so relieved it’s just user error re Italy and is doable. Thanks all 

Good Luck, I am from UK and travelling for 5 weeks this summer with my 13 year old girl. You will love it.