Andriyko

It is not really 'reclaiming Italian VAT in the UK,' though. The VAT form that you are given at an Italian store needs to be stamped by the customs when exiting the EU. If you happen to exit the EU from the UK then your form will be stamped by the UK customs to confirm that you are taking the goods outside of the EU. You then mail the form back to the address indicated and they will process the refund. I believe the refund is ultimately issued by the country where it was incurred (the administrating company claims it back), but the goods must be taken outside of the EU and it does not matter where it happens.
There are some global companies that process VAT and you may very well be able to receive a cash refund at LHR, but often these are local companies, and while the customs will stamp the form to confirm that you're leaving the EU it will not be possible to receive cash on the spot. I once claimed UK VAT, had the form stamped at TXL as that was last point in the EU, however, none of the VAT refund points at TXL worked with that UK company so I had to wait until I was back in the UK to receive cash.