With a Swiss Travel Pass, you'd need to pay for London - Paris - Basel and from Domodossola to Firenze and for your trips in Italy (often great offers for this). You'd need to have a look how much it is if bought right now for your choosen date (remember that even with a pass, you'd have to pay 30 € 2nd or 38 € 1st class for Eurostar, 10 € or 20 € for TGV in France (even 27 € 2nd or 5i € 1st class for direct TGV to Switzerland) and 10 € for italian high speed trains).
With an Eurail pass, you'd need to pay (50% reduction) for Lake Lucerne boat trip and for the cogwheel railway to Mt. Rigi (both included with Swiss Pass) and for the trips to Grindelwald and Lauterbrunnen (25% reduction with Eurail pass) (both included with Swiss Pass). The other train trips are included with both passes.
Swiss Pass also gives an reduction of 50% for the trip to Gornergrat; no reduction with Eurail. So it's 110 CHF with Eurail Pass and 55 CHF with Swiss Travel Pass (if it's already may).
With a Swiss Travel Pass, it would also be possible to do some other trips (for instance, you could do an interesting bus tour up to a small mountain village from Lake Lucerne or also go to Wengen or Mürren (recommended) from Lauterbrunnen for free).
I'd recommend the Eurail Pass for trips outside Switzerland at least. You then need to check if the Swiss Travel Pass is worth it.
What made me say that ?
The Eurostar on 3rd April is already so expensive : more than 200€ per person ! The reservations would only be 30€ but be aware that the only seats left are either on the 06:01 or 07:01 trains or after 16:22.
The TGV would cost 10€ between Paris and Mulhouse (near the Swiss border) and then you take a regional train to Basel and an Interregio/Intercity train to Lucerne (all reservation-free).
As mentioned you can also take a direct TGV Lyria to Basel but the reservation fee is higher and it only saves about 30 min.
As you plan to take trains that are not included in the Eurail pass, you should then take a look at the Swiss Travel Pass (already worth it with the Rigi + Gornergrat day trips).
So for me you should combine both passes.
The Eurostar on 3rd April is already so expensive : more than 200€ per person ! The reservations would only be 30€ but be aware that the only seats left are either on the 06:01 or 07:01 trains or after 16:22.
The OP did write "end of April", not 3rd. But as it's written: There's a need to check the prices.
Oh sorry I thought they meant April, 1 (first).
Yes, they need to check because TGV fares seem to be more expensive than usual too.
And be aware that there are track works in the Dijon area during the month of April so not all TGVs run during the day. It might be better to take the TGV from Paris-Est to Strasbourg (10€) and then a TER to Basel.
For example : leaving Paris-Est at 09:25, having a 40-min (or more) break in Strasbourg, a really nice city, taking a TER (max 200 km/h on this line) to Basel and arriving in Lucerne before 15:00.
Use the EUrail option and as above: bSMarter with how tobook and choose route.
IN the Swiss look at LOCAL passes-and in some towns HTLs may also give out free small area passes for all transit. There is certainly one for the Berner Oberland. These are much cheaper as all-Swiss Holiday pass. Looks to me as best cost-effective combo.
There is also an online bookable Swiss DAYpass, cheapest on non-busy days and booked as soon as can do-about 55CHF-valid as the HoldPass in ALL transit, also boat/bus/many mountian lines, like also Rigi =up+down other side.
