Not exactly. The reference regarding flu is https://www.independent.co.uk/news/health/flu-cases-covid-en... where apparently not a single case was detected in late 2020/early 2021, despite a period of time beforehand without lockdowns (roughly July through end of October). There are references in the media to colds being more prevalent throughout this entire same period though, but admittedly not during the lockdowns, as you say.

From this I made an uneducated extrapolation that the common cold pretty much stuck around somewhere and began to rapidly spread again as soon as people got out and about (I got one first week of schools going back in September, as per usual), whereas flu clearly did not.