Finally I found a Technet discussion on the issue, which correctly identified an incorrect registry value HPTrayCount which is set to 0 when the printer driver is changed, but is not set back to a working value.

This value needs to be changed to 0×12 (decimal 18) for each HP printer for it to work correctly. However, on print servers with lots of HP printers this can be very time consuming. To speed this process I have written the vbscript below which will check if the value exists for each printer, and if it is set to 0 will set it to 0×12.

' For each Printer in Registry check if HPTrayCount exists and is set to 0, then set to 12

'http://social.technet.microsoft.com/Forums/en/winserverprint/thread/5101195b-3aca-4699-9a06-db4578614e2d

strKeyPath = "SOFTWARE\Microsoft\Windows NT\CurrentVersion\Print\Printers"
strValueName = "HPTrayCount"
inHPValue = 18 ' Is equal to Hex 12, makesprinter driver work.

'http://www.activexperts.com/activmonitor/windowsmanagement/adminscripts/registry/#EnumRegVals.htm
const HKEY_LOCAL_MACHINE = &H80000002
strComputer = "."
Set StdOut = WScript.StdOut

Set oReg=GetObject("winmgmts:{impersonationLevel=impersonate}!\\" & _
strComputer & "\root\default:StdRegProv")

' Get list of Printers
oReg.EnumKey HKEY_LOCAL_MACHINE, strKeyPath, arrSubKeys

For Each subkey In arrSubKeys
  ' Read in HP Tray value
  strPrintPath = strKeyPath & "\" & subkey & "\PrinterDriverData"
  oReg.GetDWORDValue HKEY_LOCAL_MACHINE,strPrintPath,strValueName,dwValue
  If Not IsNull (dwValue) And dwValue = 0 Then
    'If Exists and 0 then set it to a real value to make it work
    StdOut.WriteLine subkey & ": HP Tray: " & dwValue
    oReg.SetDWORDValue HKEY_LOCAL_MACHINE,strPrintPath,strValueName,inHPValue
  End If
Next