There are many things which affect our BG. The most obvious one is food and drink. But also exercise, time of day, month, year, drugs, stress, illness, weather, ... The list seems to be endless. Unfortunately.
The illness one is your body releasing extra glucose to fight off the nasty bugs. Our bodies are amazingly clever and may detect these nasties before we are aware of them and experience any obvious symptoms. As a result, a high BG may be the first indication that you are about to become ill ... or not if your defenses are good enough to stave off the lurgy.
On the other hand some illnesses, such as food poisoning are so sudden that you don't get that build up - it's straight in (or out?) to sickness.
So when it comes to how poorly you need to be before it affects your BG, it depends.
I know that's not a great answer but when you get a sudden bout, you know and you can immediately align it to your BG levels. And when the malady is a slow burner, your BG is likely to creep up over a few days.
I can say with certainty that the worse I feel, the higher my BG goes. And the higher my BG goes, the worse I feel. It gets to the point I can't tell whether my poorliness is making my BG rise or my BG rise is making my poorliness worse. Whichever it is, I guess the treatment is the same: test, test, test and lost of correction doses.
For me, the more my BG is under control, the faster I get better.
Of course, the best thing to do is to avoid getting ill in the first place. I don't believe I have been ill more often since I have had diabetes or that any illness has lasted longer. However, managing BG rises at the same time as a runny nose, for example, is harder than just managing the runny nose so I try to avoid it.
For example, I take up the flu jab every year and, if you haven't already, I recommend contacting your GP to do so. (Although you may need to get over your illness first.)
Hope you feel better soon.