you can use Array() of worksheet names and Array() of columns to process multiple sheets:
Private Sub RMS_UpdateDateColumn()

    Dim arrWsh As Variant
    Dim arrCol As Variant
    Dim i As Integer
    
    ' the worksheets to work with, Change the name if necessary
    arrWsh = Array("Involved", "Sheet2")
    
    ' the column number where the date is located, change the column Number if necessary
    arrCol = Array(4, 4)
    
    For i = 0 To UBound(arrWsh)
        Call RMS_History(Worksheets(arrWsh(i)), arrCol(i))
    Next
    
End Sub

Private Sub RMS_History(ByRef ws As Worksheet, ByVal nCol As Long)

    Dim x      As Variant
    Dim rw     As Long, last_rw As Long
    Dim vlue   As Variant
    Dim txt    As String
    Dim dte    As Variant, tim As String

    Application.ScreenUpdating = False
    
    ' -------------------------------------------------------
    ' Set font for each worksheet
    
    With ws
        .Cells.Font.Name = "Calibri"
        .Cells.Font.Size = 11
        .Cells.VerticalAlignment = xlVAlignCenter
        .Cells.HorizontalAlignment = xlHAlignLeft
        
        ' -------------------------------------------------------
        ' Perform the basic editing
        
        ' Tidy date column by converting from text to required date format
        
        last_rw = .Cells(.Cells.Rows.Count, nCol).End(xlUp).Row
        For rw = 2 To last_rw
            vlue = .Cells(rw, nCol)
            txt = WorksheetFunction.Text(.Cells(rw, nCol), "m")
            If IsNumeric(txt) Then
                'if numeric, it is a valid date
                .Cells(rw, nCol) = CDate(Month(vlue) & "/" & Day(vlue) & "/" & Year(vlue)) + TimeValue(vlue)
            Else
                'not valid date
                dte = Split(vlue, "/")
                tim = Split(dte(2))(1)
                dte(2) = Replace$(dte(2), tim, "")
                .Cells(rw, nCol) = CDate(dte(1) & "/" & dte(0) & "/" & dte(2)) + TimeValue(tim)
            End If
            .Cells(rw, nCol).NumberFormat = "dd/mm/yyyy"
        Next
        
        
        .Columns("E:E").Delete                    ' Delete column E as this is not required
        
        ' Delete all rows with a date older than eighteen months

        .AutoFilterMode = False
        Dim FilterRange As Range, myDate As Date, myCol As String
        
        myCol = ColumnLetter(nCol)
        myDate = DateSerial(Year(Date) - 1, Month(Date) - 6, Day(Date))
        
        Set FilterRange = .Range(myCol & "2:" & myCol & .Cells(.Rows.Count, 1).End(xlUp).Row)
        FilterRange.AutoFilter Field:=1, Criteria1:="<" & CDbl(myDate)
        On Error Resume Next
        With FilterRange
            .Offset(1).Resize(.Rows.Count - 1).SpecialCells(12).EntireRow.Delete
        End With
        Err.Clear
        Set FilterRange = Nothing
        .AutoFilterMode = False
        
    End With
    
    Application.ScreenUpdating = True
    
End Sub

' chatgpt
Function ColumnLetter(ByVal ColNum As Integer) As String
    Dim vArr As Variant
    vArr = Split(Cells(1, ColNum).Address(True, False), "$")
    ColumnLetter = vArr(0)
End Function