I am about to make a trip to Britain, Ireland and Scotland from my USA home. Can I use my inReach Explorer without additional charges?

Two different answers to the two parts of the question.

As far as Iridium satellite communications go (messages, sent track points, etc.), worldwide use is governed by (and charged according to) your inReach plan, whatever it is. For example, if your plan includes x messages per month, you can send those x messages from anywhere in the world and they are still included. If overage messages are charged at y amount per message, each overage message will cost y no matter where in the world it originates.

Things like downloading maps or sync-ing more detailed (logged) track points require internet access. You are going pay for the Internet access separately, just as you do here (in the sense that the Internet access is not included in your inReach plan). This might be a substantial expense depending on the access mechanism. For example, cell phone data in the UK is unlikely to be included in your US cell phone plan.

As an aside, you should check on the availability of maps for your intended locations, particularly if you plan to download them to the Explorer+ itself. It's been a long time since I looked at it, but originally maps for locations outside of North America were limited.

Thank you, twolpert!

I would add a user name, rather than just my number, but I have not yet figured out how to do that!

Go to your profile. Click the Edit Profile blue button near the top of the right sidebar. Near the top left of the resulting profile editing page (next to the generic avatar), you'll see a box with the annoying number in it. Just change that to whatever user name you want. Can't remember if there's a save button for profile changes - look for it and use it if found, just to be sure.