"BT Infinity 1 increasing from 38 to 55 Mbps", where's the 38 Mbps still present? Ah well, I'll just get the new contract, we're up for renewal anyway.
BT Retail offer Infinity, and for this they pay BT Wholesale to resell a product. BT Wholesale in turn pay Openreach. Neither Openreach or Wholesale are stopping selling the 40Mb product, so BT Retail will have to pay Wholesale to have customers moved. It's not as simple as everyone waking up tomorrow on 52Mb.
They have to pay to have you all moved and it'll take time.
I don't see anything to indicate what they plan to do with existing users in that initial thread. They may not move you at all in the very short term.
EDIT: Incidentally your service speed is not provisioned by a guy with a laptop at the cabinet, that would be crazy. It is not an automated process, someone will have to write the appropriate scripts.
The people who actually control the equipment are not upgrading you. BT Retail have to send orders to their suppliers requesting you are all moved. In that respect it's no different from someone asking for an upgrade from 38 to 76, just if they decide to uplift existing customers they'll be doing a whole bunch of those requests.