The Denon PMA-355UK seems to be a pretty straight forward 50w/ch stereo amp. I see no reason why that would not serve you nicely -
https://www.hifix.co.uk/denon-pma355uk-black-amplifer
The only thing lacking, that might be found in a modern amp is a DAC. But we have established that a DAC, if it is indeed necessary, can be had for a very modest amount of money.
So, absolutely start with the Denon PMA-355. Then take if from there.
Your TV might have Red/White RCA connectors, but they could be Inputs, not outputs. But if it does have RCA Analog Outputs, so much the better.
Really, in my view, you just need to decide on the speakers you want.
Though the
Yamaha NSF51 are cheap, if you read the reviews on Amazon, they have generally 4 and 5 Star reviews. I think at Amazon-UK, there were 56 reviews all generally positive.
The
Wharfedale Diamond 9.1 and the
Wharfedale SW150 Sub are a bargain, both selling for considerably less than their suggest retail price because they are both long discontinued models.
The
Diamond 9.1 have a rated bass of 50hz, that should make it very easy to blend with a Subwoofer, and while £250 is a bit more than your budget, that would be a very impressive system when set up correctly.
Keep in mind, being older models, the supply on both the
Diamond 9.1 and the
SW150 is limited to available stock.
Even if you just start with the
Diamond 9.1, that is about the most speaker you can get for the least price. Spend more, you can get more, but you can't get more for a modest £100/pr.
Also note, should the
Diamond 9.1 not be available, the newer
Diamond 220 are available for a still modest £150/pr in the right finish.
I did find a place that has th
e Q Acoustic 3020 for a very low
£150/pr -
Q Acoustics 3020 Bookshelf Speakers - HiFi & Pairs - Speakers
Though the bass is only rated down to 65hz, they are very sweet great sounding speakers, and an exceptional bargain for that money.
Richer Sounds usually has some nice bargains available -
Tannoy Mercury 7.2 (6") - £150/pr -
https://www.richersounds.com/tannoy-mercury-7-2-light-oak.html
Wharfedale Diamond 220 (5"/130mm) - £120/pr -
https://www.richersounds.com/wharfedale-diamond-220-blk.html
At the
Richer Sounds price, you should definitely go for the
Diamond 220 over the
Diamond 9.1.
Cambridge Audio SX60 (6.5") - £140/pr -
https://www.richersounds.com/hi-fi/standmount-speakers/cambridge-sx60-dk-wal.html
Richer Sounds also has good prices on the
Q Acoustic 3020 - £150/pr -
https://www.richersounds.com/hi-fi/standmount-speakers/q-acoustics-q3020-matte-graphite.html
If you can come up with some serious cash, then consider -
Cambridge Audio SX80 Floorstanding (2x6.5") - £299/pr -
https://www.richersounds.com/hi-fi/floorstanding-speakers/cambridge-sx80-walnut-sos.html
At the price, the
Q Acoustic 3020 are outstanding speakers, and would very easily be able to mate with a Sub in a Stereo system.
On the Subject of Subwoofer, while you can find cheaper Subs, you can't find better value than the
BK Electronics Subs.
https://www.bkelec.com/HiFi/Sub_Woofers.htm
Even the modest
BK Gemini-II would do a good job -
http://www.bkelec.com/HiFi/Sub_Woofers/Gemini.htm
Steve/bluewizard