Calling cscript.exe and passing a vbscript to call

See if the following works

Dim objShell
Set objShell = Wscript.CreateObject("WScript.Shell")

objShell.Run "TestScript.vbs" 

' Using Set is mandatory
Set objShell = Nothing

Problem #1:

You're not seeing any output when you run your script elevated, because the RunElevated.vbs script re-launches your OSVersion.vbs script using WScript.exe. OSVersion.vbs uses Standard Input and Output which are only available from the console. Changing the command line in RunElevated.vbs to use cscript.exe instead of wscript.exe solves this problem. For more, check out my article Error Trapping and Capturing Third-Party Output in WSH.

Problem #2

It looks like your problem here is that you're mixing different kinds of quotes. A command should look like this:

"C:\path with spaces\command.exe" "parameter with spaces" "and another"

Try out these scripts.

RunElevated.vbs

' Run a script in Elevated Mode
strName      = WScript.Arguments.Item(0)
strArgs      = ""
If WScript.Arguments.Count > 1 Then
    For i = 1 To WScript.Arguments.Count - 1
        strArgs = strArgs & " " & WScript.Arguments.Item(i)
    Next
End If
WScript.echo "Running: ", strName, strArgs
Set objShell = CreateObject("Shell.Application")
Set objFso   = CreateObject("Scripting.FileSystemObject")
strPath      = objFso.GetParentFolderName(WScript.ScriptFullName)
strParams    = Chr(34) & strPath & "\" & strName & Chr(34)
If strArgs <> "" Then
    strParams = strParams & " " & Chr(34) & strArgs & Chr(34)
End If
If objFso.FileExists(strPath & "\" & strName) Then
     objShell.ShellExecute "cscript.exe", strParams, "", "runas", 1
Else
    WScript.echo "Script file not found"
End If

OSVersion.vbs

'Test passing arguments to launched script
If WScript.Arguments.Count > 0 Then
    For i = 0 To WScript.Arguments.Count - 1
        strArgs = Trim(strArgs & " " & WScript.Arguments.Item(i))
    Next
End If

' Return a string indicating the operating system
strComputer = "."
Set objWMIService = GetObject("winmgmts:" _
    & "{impersonationLevel=impersonate}!\\" & strComputer & "\root\cimv2")

Set colOperatingSystems = objWMIService.ExecQuery _
    ("Select * from Win32_OperatingSystem")

For Each objOperatingSystem in colOperatingSystems
    WScript.StdOut.Write objOperatingSystem.Caption
    WScript.StdOut.WriteBlankLines(1)
    WScript.StdOut.Write "Version " & objOperatingSystem.Version
    WScript.StdOut.WriteBlankLines(2)
Next

' Print out any passed arguments
WScript.StdOut.WriteBlankLines(2)
WScript.StdOut.Write strArgs
WScript.StdOut.WriteBlankLines(2)

' Keep the window from closing before you get a chance to read it
WScript.StdOut.Write "Press any key to continue "
strInput = WScript.StdIn.Read(1)

In order to capture the output, I would probably try a different approach using the WshShell Execute method. There are command line tools available such as Elevate or HStart that can launch command lines with elevated privileges. Here's how I would do it using Elevate.

' Run a script in Elevated Mode
strName      = WScript.Arguments.Item(0)
strArgs      = ""
If WScript.Arguments.Count > 1 Then
    For i = 1 To WScript.Arguments.Count - 1
        strArgs = strArgs & " " & WScript.Arguments.Item(i)
    Next
End If
WScript.echo "Running: ", strName, strArgs
Set WshShell = CreateObject("WScript.Shell")
Set objFso   = CreateObject("Scripting.FileSystemObject")
strPath      = objFso.GetParentFolderName(WScript.ScriptFullName)
strParams    = Chr(34) & strPath & "\" & strName & Chr(34)
If strArgs <> "" Then
    strParams = strParams & " " & Chr(34) & strArgs & Chr(34)
End If
If objFso.FileExists(strPath & "\" & strName) Then
    Set WshShellExec = WshShell.Exec("elevate cscript.exe " & strParams)

    Select Case WshShellExec.Status
        Case WshFinished
            strOutput = WshShellExec.StdOut.ReadAll
        Case WshFailed
            strOutput = WshShellExec.StdErr.ReadAll
    End Select
Else
    WScript.echo "Script file not found"
End If

WScript.echo strOutput