the River Neva/the Neva River ??

[Verona_82]

Hello!

I'd love to find out if there are any rules regarding the use of geographical names of rivers along with the common noun in a sentence. For example, what should I say:

... on the River Thames (1)
or ....on the Thames River? (2)

We say Lake Ontario, Mount Everest, but the Aegean Sea. When I googled the phrase, the majority of hits were in favour of (1). However, I found out there were different alternatives for 'the Seine":

on the river Seine
on the Seine river

and there're four alternatives for the Neva , the river that my hometown is on. Of course, Google doesn't have a monopoly on truth, but I'm really eager to avoid any mistakes when speaking about it.
I'd be grateful if anyone could shed some light on this.

[5jj]

... on the River Thames (1)
or ....on the Thames River? (2)

#1 is the normal order, though you may see #2 when it is used adjectivally: Thames River Police.

[SoothingDave]

#2 is normal in America. The Mississippi River.

[Verona_82]

Thank you!
There're no rules

Does that mean that
-The city is on the River Neva would be preferred in BrE

whereas

The city is on the River Neva/the Neva River would be ok in AmE?

[5jj]

-The city is on the River Neva would be preferred in BrE

Yes, but if it suddenly appeared in news reports as the Neva River, that's what it would be.