If you love your Cheviots you're perhaps unlikely to get on with the Naim sound. Worse IME Naim and vintage Tannoy are a truly unhappy combination that brings out the best in neither. The Tannoys want to be rich, fluid and a little loose, Naim will feed them hard dry cardboard. And will drain your bank account into the bargain.
If you like vintage gear you could always try other seventies receivers or amplifiers. Many sound very good, there's lots of choice and they nearly always have good phono stages.
Your Cheviots are very efficient speakers so there's no need to chase the big expensive TOTL models. This is a huge advantage for you because the smaller versions usually sound the same but can often be had for £50 or less. And there are loads to choose from because they sold in such vast numbers. If you really are considering a Naim Atom then you can afford to experiment here and you'll learn far more buying and running a few examples that than you will trying to use forums to shortcut a selection process that is unavoidably personal and subjective. You can always sell on those you don't get on with.
If you're not keen on the Sansui house sound there is every chance you'll find another you do like. JVC, Hitachi or Pioneer might suit you. Or, if clarity is your thing, for a little more a Yamaha CR620 or even a CR820 might be had for or a hundred pounds or so. An extra ace up the sleeve of this particular range is sophisticated tone control (including the fantastic variable loudness knob) which might help you get a sound you like. Though I haven't myself tried Yamaha's with Tannoys. Someone has to be first
A more leftfield and very stylish choice might be the Norwegian brand Tandberg. I've found they work quite well with vintage Tannoys. Not the last word in clarity or neutrality of course but very dynamic. They squeeze more bass and more life from big Tannoys than many other amps manage. And their radios are up there with the very best. Try the TR1000 if that appeals.
As for streaming there are a bunch of good ways of doing this. You could get a bluetooth DAC like the iFi Zen Blue (£160) . and simply stream from your phone using whatever apps you like. My daughter does exactly this into a seventies receiver and loves it. Or buy a second hand usb dac (say a Dacmagic at £90) and drive it from a laptop. Or assemble your own streamer from raspberry pi stuff.
There are a bunch of bad ways of streaming too and buying an Atom (or any other factory made streamer) would be one of them.
The problem with factory streamers is not the sound (they usually sound as good as the options above) it is that they are pointlessly expensive, they date very quickly, you can't change the user interface and you're reliant on the factory for software upgrades and repairs.
The simple fact is that equally good sound can very easily be had for far less £££s and in more flexible, adaptable longer lasting packages.