Which country's passport will you use?

No, it won't change that. US citizens are not going to need visas for the UK after Brexit. The Brits already shot themselves in the foot with Brexit. They're not going to make it worse by introducing a visa for US passport holders.

WHY do you think there would be any effect on you?? It is an issue between the UK and the EU.

Have no idea what BDKR's post is about - just his way of inserting politics into everything

I'm not inserting politics into everything. The vast majority of my posts have nothing to do with politics. but the title of the topic is Brexit. If that's not politics than what is it?

"WHY do you think there would be any effect on you?"

They know that they don't need a visa to any EU member state and I can understand they're wondering if that's going to change with the UK leaving the EU.

>>They know that they don't need a visa to any EU member state and I can understand they're wondering if that's going to change with the UK leaving the EU.<<

And WHY would that be???? The do not need visas to visit the UK now so why should they later???

With regard to short stay/ tourist entry regulations, the UK already has a different regime than the Schengen area - which all of the other countries you mentioned are part of.
Your visa free entry to France, Germany, Switzerland and the Netherlands is not affected by Brexit anyway.
I'd say that it will be highly unlikely that the UK changes the visa free entry for US citizens as there is no reason why they should do so in the first place, and it would also impact UK citizens travelling to the US as these visa free / visa waiver schemes rely on reciprocity.

"And WHY would that be???? The do not need visas to visit the UK now so why should they later???"

It may be natural for us, but its not obvious for every traveller. I remember not so long ago even many Brits were wondering if they would need a Schengen Visa after Brexit! My answer was that if citizens of countries like Paraguay, Nicaragua and the Solomon Islands have visa-free access to the Schengen Area, than UK citizens probably won't need one either.

Unless all the countries you mentioned decide to change the rules of entry between now and whenever, no.

BDKR - stop muddying the waters -- the OP is from the States. Paraguay et al have no relevance to anything. And what Brits may or may not have to do later also has no bearing.

The only possible worry is a longer check when you travel between the UK and France. Still likely shorter than any customs and immigration check you'll face returning home.

As BDKR wrote, yes.

There is, but it affects only EU citizens.
We can travel to every member state without applying for visa and without stating any reason why we want to go there.
If there is border control, e.g. as between the Schengen countries and the UK or Ireland, the agent checks my identity - usually scans my passport or ID card. Or just looks at the document and makes a visual check. That's it.
Once the UK has left the EU, there will be some other regime regulating the short-term visits of EU citizens to the UK and vice versa.
As a citizen of any country which is not an EU member state and which is not the UK, all of the above is totally irrelevant for your travel plans.
It could take a bit longer to clear passport control when (in your case) arriving from the UK in France, but there can as well be mitigating measures like increasing number of staff so there won't be any delays. Who knows.

The UK has never been a Schengen country anyway, so visas have nothing to do with Brexit. The Schengen Agreement, which includes several non-EU members, is what governs immigration. US passport holders have always been allowed entry into Britain as tourists without a visa even before it joined the EU.

Nothing changes for anybody until the end of 2020 anyway. Europeans can still settle in the UK and vice versa. EHIC cards remain valid, etc etc.

From January 1st 2021 all non EU/Schengen/EEA visitors will need an ETIAS visa waiver (like the ESTA, but cheaper and valid for longer) to visit all EU countries and those that are members of Schengen but not the EU. The UK is talking of something similar for visiting the UK.

https://etias.com/

"From January 1st 2021 all non EU/Schengen/EEA visitors will need an ETIAS visa waiver"

Again, a stupid, useless and counterproductive rule by the brainamputated bureaucrats of Brussels. No wonder so many Brits had enough of this flea circus. I pity them, but can't blame them.

This is only making it more difficult for genuine visitors like tourists, businessmen, students to come to the EU. Isn't it ridiculous that an Indian or Russian family who want to spend their money in Europe for travelling have to go through all kind of hassle to get a visa(if they get it all), while NGO ships shuttle scores of African immigrants to Europe without any ETIAS, Schengen Visa, passport or anything?

Is the visa wavier really anything different from what the US government currently requires of EU visitors to the US?