Bottom line, you need appliances that work on 220. They need to have plugs that will work in the receptacles of the countries you are visiting.
You are going to England and to Italy. Both are 220, but the plug receptacles are different.
The easiest option is to use the hairdryers provided by your lodgings. I've never stayed anywhere in Europe that did not have a hairdryer. I've never used one there that didn't dry my hair.
Appliances that straighten hair are not provided. Try styling your hair without a flat iron before you go. You may find a way to wear it that doesn't require extra equipment.
If you find that you can't work that out, buy one
here or in the country of your arrival, and get the appropriate plug adapter(s) for it.
The simplest plug adapters from US plugs to those used in Italy are this type. I've been using them for years. As the description says, this type doesn't work in the UK or Ireland, so you need one like this for England.
If you buy a hair straightener device in the UK, you will need a plug adapter like this one for Italy. If you buy one in Italy, you'll need one like this for the UK. I'm sure that if you buy this appliance in the UK, you'll be able to get a plug adapter for Europe and vice versa.