I have the same problem. In my case I have a Hub5 router and Envy 5020.
Used to work, though frequently claiming printer offline.
I can access the printers web server, print documents and HPsmart claims all is well. The only problem being connecting to the printer for ink monitoring.
Yup, that’s exactly the situation I had...
I chatted with the support guys through online chat, and the solution offered was to replace the HH5 with a Smart Hub 2; I’ve had no problems with Instant Ink since switching. Along with some others, however, I am having difficulty using AirPrint and smart devices (plugs, etc.) which has been attributed (in another BT Community thread) to new/current Smart Hub 2 firmware, so if your solution ends up the same be prepared for the possibility of some other problems.
Wish I had a ‘change this setting to make it work’ solution to offer, but alas not!
It may or may not be of help.
I have a HP Envy Pro 6400 series with instant ink sub. I do not use a BT HomeHub. I had this issue a couple of months ago and the problem was with HP not BT. I called the HP help line and after two attempts the problem was discovered at their server end. They reset my account and all has worked fine since.
Hi, I have a HP OfficeJet Pro 6960 connected to Smart Hub 2. I was able to open the HP Smart app but had problems opening the Ink supply details. Seems there was an update on IOS on 21 April and I now have no problem with access. So I am presuming it is at the HP end.