You are presumably using Outlook without the option to send files immediately so they are accummulated in the outbox, where you are processing them. The problem with this is that there may be other unrelated e-mail messages pending in the outbox that would screw up your process.

The easiest way to keep the Word window out of the way is to minimize it

wdApp.Application.WindowState = 2
Do this after declaring wdApp immediately before 
On Error Goto 0
Dialogs(80)
is exactly the same command as 
Dialogs(wdDialogFileOpen)
but the numeric equivalent is used to allow late binding to Word (there is no need to set a reference to Word in Tools > References when using the code I posted).

The probable reason your GetFile function doesn't work for some is that they don't have Word declared as a reference in their Outlook VBA editors. This is why I prefer to use Late Binding to other office apps. It is harder to program with late binding, but it improves compatibility. GetFile is the better option so I have modified it to work from my version of the original code. It doesn't need yet another instance of Word as you already have Word opened, so use that one.

Option Explicit

Sub SetCCandattach()
Dim wdApp As Object
Dim Maillist As Object
Dim Datarange As Object, Datarangecc As Object
Dim i As Long, j As Long
Dim Item As Outlook.MailItem
Dim Attachment As Attachment
Dim olNS As Outlook.NameSpace
Dim MyFolder As Outlook.MAPIFolder
Dim count As Integer
Dim strFileName As String
Dim bStarted As Boolean
Const strPath As String = "C:\Path\" 'The initial filepath for the maillist document

    ' This sub assumes that this macro is being run from within Outlook
    On Error Resume Next
    'Set OutlookApp = GetObject(, "Outlook.Application") 'You are already in Outlook?

    ' Open the mailmerge table document

    Set wdApp = GetObject(, "Word.Application") 'Get Word if it is already running
    If Err Then
        Set wdApp = CreateObject("Word.Application") 'or create it if it isn't
        bStarted = True
    End If
    wdApp.Application.WindowState = 2 'Minimize Word
    On Error GoTo 0
    strFileName = GetFile(wdApp, strPath) 'Get the maillist filename
    If strFileName = "" Then
        MsgBox "Process cancelled"
        GoTo lbl_Exit 'User cancelled
    End If
    
    Set Maillist = wdApp.documents.Open(strFileName) 'Open the selected document
    ' iterate through all items in the Outlook Outbox and adds an email from column 2 and an attachment from paths in colums 3plus


    ' By following change eliminates the security access prompts!
    'Set olNS = oOutlookApp.GetNamespace("MAPI")
    Set olNS = GetNamespace("MAPI")
    Set MyFolder = olNS.GetDefaultFolder(olFolderOutbox)
    For i = 1 To MyFolder.Items.count
        Set Item = MyFolder.Items(i)
        Set Datarangecc = Maillist.Tables(1).Cell(i, 2).Range
        Datarangecc.End = Datarangecc.End - 1
        Item.CC = Datarangecc
        For j = 3 To Maillist.Tables(1).Columns.count
            Set Datarange = Maillist.Tables(1).Cell(i, j).Range
            Datarange.End = Datarange.End - 1
            If FileExists(Trim(Datarange.Text)) Then
                Item.Attachments.Add Trim(Datarange.Text), olByValue, 1
            End If
        Next j
        Item.Save
        Item.sEnd
        count = count + 1
    Next i
    Maillist.Close 0
    MsgBox count & " emails have had a cc added."

    'Clean up
lbl_Exit:
    If bStarted Then wdApp.Quit
    Set Maillist = Nothing
    Set wdApp = Nothing
    Set Datarangecc = Nothing
    Set Datarange = Nothing
    Set olNS = Nothing
    Set Item = Nothing
    Set MyFolder = Nothing
    Exit Sub
End Sub

Private Function FileExists(filespec) As Boolean
Dim fso As Object
    Set fso = CreateObject("Scripting.FileSystemObject")
    If fso.FileExists(filespec) Then
        FileExists = True
    Else
        FileExists = False
    End If
lbl_Exit:
    Exit Function
End Function

Function GetFile(OtherObject As Object, strPath As String) As String
Dim sItem As String
Dim fdFile As office.FileDialog
    Set fdFile = OtherObject.FileDialog(msoFileDialogFilePicker)
    With fdFile
        .Title = "Select a File"
        .AllowMultiSelect = False
        .InitialFileName = strPath & "*.docx"
        If .Show <> -1 Then GoTo NextCode
        sItem = .SelectedItems(1)
    End With
    GetFile = sItem
lbl_Exit:
    Set fdFile = Nothing
    Exit Function
NextCode:
    GetFile = ""
    GoTo lbl_Exit
End Function