I’ve been waiting for my supporting documents to be returned for five weeks now (10 since I mailed them). I’ve called several times – told two weeks, then three, then four…. Still waiting. I’m not sure I’ll ever see them again, to be honest. This system is a shambles.
Is your travel all within the EU / EEA, and do you have Settled Status? You can update your Settled Status to be linked to your ID card, and then travel on that. I did this, and I had no issues at the U.K. border. I had more issues with airlines and with entering / exiting Schengen, but I was able to talk my way through them.
The bigger problem for me was the loss of my US Passport. The US will not allow me to enter on a foreign passport. So after five weeks of this nonsense, I had to go to the embassy and get an emergency passport, at a cost of $165, which resulted in the cancellation of the passport that HMPO still has. The embassy told me that this happens all the time, and they’re very frustrated with the British losing American (and presumably other) passports. I was the second person with this issue at a 7:45 am appointment.
I’ll have to then exchange this passport for a permanent one within the next year. The knock-on effects of HMPO’s incompetence are endless.
To be honest, I regret getting the British passport as I could have just continued using the European one indefinitely. If I were you, I’d wait until thing settle down (definitely after summer). Hope this helps.