Hi and welcome to the forum.
I don't know much about Hashimotos other than it is another autoimmune condition and if I remember correctly is treated with steroids? Hopefully someone will correct me if I am wrong. As a Type 1 I don't find I am prone to more illness than I was before and indeed up to last year I actually had far less colds and viruses than I did before diagnosis 6 years ago, but January and February this year I made up for it and had 2 in a month. The first was relatively normal and I powered through it but the second a couple of weeks later took me a good 3 weeks and my breathing was affected as well being snotty for ages. I suspect that second one may have been Covid but I was out of Lateral Flow tests, so I just kept myself to myself until it had gone.
The important thing with diabetes is to manage it as well as you can and particularly through the illness to help your body recover quickly because high BG levels are likely to foster bacteria and potentially lead to secondary infections. Did you find that your BG levels increased when you were ill or are you one of those people whose levels decrease during illness? Have you done a DAFNE course and do you have sick day rules which consist of a flow chart to help you manage your levels more effectively during illness?
Generally I need to increase my insulin and I increase both my basal and bolus but I have one of the shorter acting basal insulins (Levemir) which gives me the flexibility to increase and decrease on a more or less day by day basis, whereas other longer acting basal insulins like Tresiba need 3 days to fully effect any changes in dose and can leave you dropping low when you come to the end of your illness and need to reduce the dose back down, so I think some people just manage their high levels with corrections of fast acting insulin until they are well again.
As I said, I am not sure how Hashimoto's would impact your immune system and if it could potentially make you more vulnerable to catching viruses and fighting them off once you have caught one. I think
@Pumper_Sue has Hashimotos and Type 1, so may have a better insight but she doesn't post very frequently on the forum these days.