Mobile plan for Austria and Northern Italy?

Yes. You have a few options:
- get US phone service before you leave that includes international roaming (e.g. T-Mobile)
- buy a European SIM card before you leave the US that you can activate and prepare before you land in Europe
- buy a European SIM card when you get to Europe.

There are pros and con of each approach. I use a Dutch Vodafone SIM I bought (unopened) on eBay a few years ago and have used it in Slovenia, Italy, France, Spain, and Portugal (and would work in Austria) on several trips. Today I'd probably buy a UK Vodafone or Three SIM ahead of time. Search the forums for "UK Vodafone" to find the long recent thread about how to get going with a UK Vodafone SIM.

If you are landing in Italy first, you might have a better chance getting a prepaid SIM that works in Italy than if you land in Austria first. In theory, all European SIM cards are supposed to support free roaming in other European countries now - but in practice, there are a lot of restrictions and exceptions.

More info about Italian and Austrian SIM cards here:

https://prepaid-data-sim-card.fandom.com/wiki/Italy
https://prepaid-data-sim-card.fandom.com/wiki/Austria

IS there such a thing as a "cheap smart phone" ?? I use my everyday Samsung 6 and switch out to a SIM from Vodafone NL -- and I have used it in Austria.

Orange in France sells a house brand ‘smart phone’ for 80€! We’ll be there
in April so maybe I’ll grab one of those.

We change planes at Heathrow so I thought about getting a UK Vodaphone.
Only looked at their site briefly but it appeared that using it outside UK would
be expensive.

I thought the EU was making it illegal for inflated roaming charges? That would
mean I could get an Orange plan in April and not worry about it a few months later
in Austria and Italy. It would probably be cheaper to get a 30 day SIM in April and
buy another in Austria.

Have you already read this recent thread regarding U.K. Vodafone SIM?

https://community.ricksteves.com/travel-forum/tech-tips/vodafone-uk-sim-card-for-use-in-europe

I have used one that I purchased in a Vodafone shop in London and have been able to roam in Germany and Switzerland

The Vodafone UK SIM to which Andrew H refers above is inexpensive, including for usage outside the UK. I bought and used my SIM last fall for 3.5 weeks (all outside the UK) for $16.84. At that time, it included 3GB data (which they doubled) with 250 minutes and unlimited texts. The most recent deal I saw on their site was triple data for their £10 "Big Value Bundle" for use across the EU and the UK.

We change planes at Heathrow so I thought about getting a UK Vodaphone.
Only looked at their site briefly but it appeared that using it outside UK would
be expensive.

Outside of Europe it would probably be expensive, but it would be great for Austria and Italy - no roaming charges at least for data. If you need to make local calls within Europe, that might be extra. I've only used the Dutch Vodafone SIM so not sure of the calling restrictions with the UK version, but I don't need to make many calls within Europe while traveling. (If I did, I'd install WhatsApp, as so many locals in Europe already use that, and that is free.)

You can certainly buy the UK Vodafone SIM ahead of time before you leave the US and not have to stress about trying to buy one while changing planes. Here's the detailed thread about doing that:

https://community.ricksteves.com/travel-forum/tech-tips/vodafone-uk-sim-card-for-use-in-europe

IS there such a thing as a "cheap smart phone" ?? I use my everyday Samsung 6 and switch out to a SIM from Vodafone NL -- and I have used it in Austria.

I guess it depends what you mean by "cheap." My everyday US smart phone is a Moto X4 that I paid $165 for (free and clear, not through any carrier), and that's probably a lot cheaper than most smart phones folks here use. It's been a great phone. There are some cheap unlocked Android phones that can sometimes be had for closer to $100 outright.

Thanks to all! Found a new in box Nokia 6 on Amazon for $110!
Vodaphone has a 30 day SIM with 20GB data and unlimited minutes and texts for £22!
The £17 option is same but for 5GB data which should be plenty for a month, eh?
And BTW roaming is free anywhere in Europe!