This is the HMRC info on VAt refunds. The link you gave looks to be for Schipol rather than LHR

You have to get customs to stamp the forms etc and then get the actual refund from Travelex. Travelex only pay you the cash (and take a fee). I don't think it makes a difference if you do this part airside rather than landside.

Here is a Travelex guide

The guidance says that you need to show the goods to customs which would be hard to do if they are in your checked baggage and you use the airside customs desks.