International charges on my phone while traveling in Europe

Put the phone in "aircraft mode", but enable WiFi.
That way no incoming or outgoing calls, data, or anything else. But you will still be able to use the internet and internet based services so long as you are connectect to a WiFi hotspot.

If you just turn "data roaming" off, you would incur no data charges, but you would still be able to make outgoing calls, and other people would still be able to ring you, both at your cost for the international leg.

To avoid international charges, I put my phone in airplane mode and then turn on WiFi. WhatsApp needs an internet connection - whether through cellular connection (charges) or WiFi (free). You can download some Google maps and Google translate dictionaries to use off line so you don't need any internet connection, but they do take up storage space. As far as I know, you cannot get your regular text messages over WiFi unless you have an iPhone (iMessaging). If you get a sim card in London, you will have to turn cellular back on, but it will not be associated with your phone number (because your sim card would no longer be in the phone). A foreign sim requires your phone to be unlocked by your carrier - if you're not sure, call before you rely on this option.

You could consider getting an international plan just to be on the safe side. Most major carriers offer a per-day option (you only pay if you use the phone to make calls) but the data rates are VERY expensive, so don't browse the internet or use Google maps unless you're on WiFi.

You can get non-iPhone text messages over WiFi but it's temperamental. I have a Samsung Galaxy with Verizon, which I kept on airplane mode for the entire 16 day trip. I'd go days without getting texts then all at once I'd get a batch that had been sent over several days. And there was no way to tell when they were actually sent.

Whatsapp, google translate, google maps or browsing the internet use "data" - this can be wifi or cellular. If you turn off cellular data then they will only work when you are connected to wifi. Cellular data is the thing you will be charged for.

Regular texts and calls go over the cellular network but are not "data" so will still work with cellular data disabled. You will probably be charged for receiving texts though (with an American plan at least.)

If you get a UK SIM none of the above changes but your phone number will. The UK SIM will probably have cheap data rates though so you might not want to bother turning off data.

Finally I don't think you can phone Uber, you use their app, which takes you back to using some kind of data.

Who is your provider?

Here's a good article from this forum that may help. cellular vs airplane mode

As far as voice mail goes, I hate voice mail, so I let it fill up where no one can leave a message. You either get me or you don't.
I do turn off ALL app and email notifications AND data on my phone prior to leaving the country. You want notifications off so just in case you should have to turn data on for an emergency all of your apps and email don't start updating and wham you with a big data bill. If you do turn it on don't forget to turn data back off as soon as you are finished! I generally keep data turned off on my tablet unless traveling in the US somewhere I don't get a wifi signal.

For what it's worth, we traveled five weeks this winter on wifi only. It took a little planning on my part in case we found ourselves in places with no wifi. I had a windows phone so many great apps available to android and iphone weren't available to me but I still found city maps to download for offline use. Many people here love CityMaps2go for android and apple. GPS doesn't need data so I could see where I was when using the apps. For train schedules and such, I looked them up on my tablet using wifi and then took a snap shot with my phone to refer to. We did fine using whats'app and facebook messenger to communicate with folks back home and in Europe. I was planning on doing that again this fall but will have to use the Verizon $10/day plan one or two days so someone on our trip can take advantage of a VRIF happening right in the middle of our trip.

If you get a sim to use in London and Paris then all of this is moot. From what I have seen, several providers offer excellent plans with voice, text, and minimal to gobs of data. Make sure you put your original sim in a very safe place so you can reinstall when coming home.

For google maps you can download maps while on wifi to use offline. I haven't tried it but others have reported here in the forums that it works well.

If you get a local SIM and plan with a data allowance there is no need to turn off data (unless you're worried that background app refreshes or you otherwise use up cellular data inadvertently).

I was in London earlier this year on my way to Switzerland. The very helpful staff at the Vodafone store on the Strand opposite Charing Cross station set me up. I got a SIM and a plan with U.K. calls, texts and 1GB. of UK data and 500 MB of roaming data. The roaming data was valid in Switzerland and many EU countries. I believe it was £17 and valid for 30 days. It was called the big value bundle.