Object Variable Or With Block Not Set

Option Explicit will not stop errors, it helps prevent you making them
Also, look at your code you needlessly switch off screen updating twice. You switch off displaalerts but don't restore it

It looks like your error handling was clicking in if the sheet isn't found. Never use error handling until the code is working

See if this is better. I've tidied the code up

Option Explicit
Public Sub Alex()

    Const strFldrPath As String = "C:\Data\Communications Division\Testing Folder\Working Time Master\Working Time Records\"
    Dim wb As Workbook
    Dim ws As Worksheet
    Dim Filecell As Range
    
'On Error GoTo err_trap '< remove ' to restore error handling
    With Application
    .ScreenUpdating = False
    .DisplayAlerts = False
    
       
    For Each Filecell In Intersect(ActiveWorkbook.Sheets("Sheet1").UsedRange, ActiveWorkbook.Sheets("Sheet1").[A:A])
        If ActiveWorkbook.ReadOnly = True Then 'Why ????
            ThisWorkbook.Sheets("Sheet2").Cells(This 'Why?Workbook.Sheets("Sheet2").Cells(Rows.Count, 1).End(xlUp).Row + 1, 1).Value = ActiveWorkbook.Name
            ActiveWorkbook.Close False
        ElseIf Dir(strFldrPath & Filecell.Text) <> vbNullString Then
            Set wb = Workbooks.Open(Filename:=strFldrPath & Filecell.Text)
            ws.Copy After:=wb.Sheets(wb.Sheets.Count)
                   
            For Each ws In ActiveWorkbook.Worksheets
                If ws.Name = "20 DEC 10 - 20 MAR 11" Then
                .DisplayAlerts = False
                ws.Delete
                .DisplayAlerts = True
            End If
          Next ws
        End If
      ActiveWorkbook.Close True
    Next Filecell
err_trap:
    .DisplayAlerts = True
    .ScreenUpdating = True
  End With
On Error GoTo 0
End Sub