Change the Formula in E2 to = "=A2" and copy down.

Change the Shift Function to 
Function Shift(D As Range, DayShift As Boolean) As Long
Dim T As String
    T = Format(D, "hh:mm")

    If DayShift Then
        If "06:00" <= T And T < "22:59" Then
            Shift = 1
        Else
            Shift = 0
        End If
    
    Else
        If ("23:00" <= T And T < "23:59") Or ("00:00" <= T) And (T < "06:00") Then
            Shift = 1
        Else
            Shift = 0
        End If
    End If
End Function
You can Format Cells in Column E to only display times.

This won't work for any old Data sheets

You can use a Formula to call this function in Column E of old Data sheets to retrieve the date from Column A 
Function GetDate(Dispatched As Range) As Date
    GetDate = Split(Dispatched, " ")(0) & " " & Split(Dispatched, " ")(1)
End Function
E2 = GetDate(A2)