How to detect a value change of a Datepicker using SwiftUI and Combine?

Please find below one possible approach (with a bit modified your code). Here is an idea to have source state transparent to activate dependent within it. Hope it will be helpful somehow.

Here is demo

Here is code

struct TestDatePicker: View {

    private static func weekOfYear(for date: Date) -> Double {
        Double(Calendar.current.component(.weekOfYear, from: date))
    }

    private static func weekDay(for date: Date) -> String {
        let dateFormatter = DateFormatter()
        dateFormatter.dateFormat = "EEEE"
        let weekDay = dateFormatter.string(from: date)
        return weekDay
    }

    @State private var date: Date
    @State private var weekOfYear: Double
    @State private var weekDay: String
    @State private var lastWeekOfThisYear = 53.0

    private var dateProxy:Binding<Date> {
        Binding<Date>(get: {self.date }, set: {
            self.date = $0
            self.updateWeekAndDayFromDate()
        })
    }

    init() {
        let now = Date()
        self._date = State<Date>(initialValue: now)
        self._weekOfYear = State<Double>(initialValue: Self.weekOfYear(for: now))
        self._weekDay = State<String>(initialValue: Self.weekDay(for: now))
    }

    var body: some View {

        VStack {

            // Date Picker
            DatePicker(selection: dateProxy, displayedComponents: .date, label:{ Text("Please enter a date") }
            )
            .labelsHidden()
            .datePickerStyle(WheelDatePickerStyle())

            // Week number and day
            Text("Week \(Int(weekOfYear.rounded()))")
            Text("\(weekDay)")

            // Slider
            Slider(value: $weekOfYear, in: 1...lastWeekOfThisYear, onEditingChanged: { _ in
                    self.updateDateFromWeek()
                })
            }

    }

    func updateWeekAndDayFromDate() {
        self.weekOfYear = Self.weekOfYear(for: self.date)
        self.weekDay = Self.weekDay(for: self.date)
    }

    func updateDateFromWeek() {
        // To do
    }

    func setToday() {
        // To do
    }

    func getWeekDay(_ date: Date) -> String {
        ""
    }
}

struct TestDatePicker_Previews: PreviewProvider {
    static var previews: some View {
        TestDatePicker()
    }
}