Did you get a combined jab or 2 separate ones ?
At the same time ? Don't fancy 2 sore arms as I only have 2 !
I bought mine privately, because of availability and location/logistics this autumn (there isn't a local pharmacy offering covid vaccine privately, whereas private flu is relatively easy to find) these were 4 weeks apart.
Two years ago when I was eligible for NHS provided covid vaccine I had flu and covid vaccination within 2 days of each other, one in each arm. I actually preferred this because it got it out of the way in a couple of days, rather than sore arm 3 days one week, sore arm 3 days a few weeks later.
However this year I hardly had any soreness from the flu vaccination, the covid obtained privately a few days ago was very sore but now easing so very relieved.
I think it's easy to over think this, and so I just tend to go with availability.