Who's responsible for notifying a time flight change?

A full-service carrier may go to some lengths to keep passengers informed— pre-merger United would even call me on the phone to confirm schedule changes if I did not acknowledge an email within a day or two— but with a no-frills carrier, I would have been checking and re-checking everything constantly with the airline in the week before the flight.

The eDreams Terms and Conditions lay out that they are only an intermediary, and that your primary agreement and responsibilities are with the airline, even explicitly saying to check the terms and conditions set forth by the airline, and

Further to ensure that You have provided correctly all passengers’ personal data during the online booking process, we recommend to check the airline’s policy on this.

According to the TUIfly general conditions of carriage, i.e. the contract you agreed to when you purchased the ticket:

10. Flight schedule and cancellation of flights

1. Flight schedule. The expected times of departure and arrival are always indicated in the local time of the relevant airport. The carrier shall do everything within their power to transport passengers and their baggage swiftly and efficiently. The carrier will make every effort to respect the announced flight schedule. When you book your ticket, we will inform you of the flight schedule applicable at that moment. It is possible that the flight schedule will change after you have made your booking due to circumstances beyond our control (decisions by competent authorities or weather conditions). If you gave us your contact details, we will inform you of any changes. By booking a flight with TUI, you acknowledge that the exact flight schedule is not essential for you. However, if we make a significant change to the flight schedule after you have made your booking which is not acceptable to you and if we are unable to offer you an acceptable alternative under similar conditions, we will refund your ticket. This provision does not affect the rights of the traveller pursuant to Regulation (EC) N° 261/2004 establishing common rules on compensation and assistance to passengers in the event of denied boarding and of cancellation or long delay of flights.
2. Verification of the flight schedule. You can also at all times consult the actual flight schedule on our website by following the procedure described in your travel documents.
3. Cancellation. If a flight is cancelled, or if a booked seat which had been confirmed earlier is not available, the carrier will always reimburse the price of the ticket should the passenger concerned refuses alternative transport under comparable conditions. Furthermore, the carrier will award additional compensation, insofar said compensation is payable in accordance with the applicable legislation and regulations, including EU Regulation 261/2004.

(emphasis added. Note that this in and of itself may not absolve TUIfly of legal or regulatory requirements, but I'm not aware of any legal resource you have here.)

eDreams probably attempted to pass on your contact information, and TUIfly probably attempted to contact you using whatever contact information they had for you. But you don't have an agreement that obligates either of them to confirm this contact information— the onus is on you, and to double-check in advance of the flight that it will operate as scheduled, just as the onus is on you to meet baggage restrictions or document requirements.

This is not to say you should avoid third party bookings. I book plenty of flights on American Express or on Orbitz, for example. But if booking through a third party, I would always advise you to look up your reservation in the airline's system after purchase to confirm everything there, and to contact the airline as soon as possible through appropriate channels if anything is amiss. I sometimes have problems with my name (first and middle names stuck together as one, ahem Chase Sapphire; or Mr. attached to the first name, ahem OneTravel), often have problems with the right frequent flyer number being entered (a corporate CWT portal still asking for my "Continental" number), and almost always have problems with seat preferences. I certainly don't trust that my phone number and email address are always being sent along unmolested.