I think by 'BT do something' he was referring to (BT) Openreach DLM.
To rento: DLM continuously monitors the line and adjusts sync speed according to what it sees; in particular how many errors there are. However, during the first 2 days or so it monitors but only makes adjustments if it sees severe errors. During those first two days the sync is as fast as it can manager (almost) regardless of errors. It is typical that the speed will lower a bit about two days in.
I agree that a change as soon as possible is probably the best. That way DLM will adjust your line speeds appropriate to what your modem can manage rather than the modem in the HH5. It won't matter too much, as it will adjust (maybe more slowly) anyway. once yours is plugged in. There could be a small advantage of keeping the HH5 for a few days so that if there are any issues you have 'standard' equipment which will help in discussions with BT service. Hopefully you won't have any issues anyway.
Be careful to power off any modem, or if possible shut it down, before disconnecting the telephone. Otherwise DLM is more likely to see the interruption as an error.
Recently moved from BT Infinity 2 to PlusNet. Very happy so far.