£300-£400 will buy a respectable stereo amp and speakers. However, you're unlikely to get a subwoofer output on the amp at the lower end of your budget unless you (or he) desides to sacrifice audio fidelity for features.
To keep both, the Yamaha AS301 amp @ £225 is a good option. You also get at least two digital audio inputs (optical and coax) which means direct compatibility with a TV, Sky/Virgin/Freeview box if they're his TV sources, and that's without spending anything extra on adapter boxes or external DACS as you might have to with more budget-oriented main brand amps. Having all-in-one in a quality amp saves money and keeps things simple too.
For speakers, one of your or his concerns is bass from small bookshelf speakers. The answer to that is simple: Don't buy bass-lite small bookshelf speakers. Lol
If you'd have been making some decisions 6 months ago, then my go-to recommended choice for a standmounter with decent bass was the Monitor Audio Bronze 2. That's because they were big, well-made, bass-ported standmounters. If I think back to starter Hi-Fi in the early '80s, this size of speaker would have been your typical 1st purchase. Tens of thousands of people listened to their music for years on stuff like this without having serious thoughts about adding a sub.
In the mid-80s the trend started for smaller speakers. It was a compromise to price. Wharfedale Diamonds did music just fine at a little over half the price of bigger alternatives just so long as you could live with the bass-lite presentation.
The trend for small continues, and due to inflation, small speakers are incredibly cheap compared to their equivalents from 30 years ago. What hasn't change though is physics.
Make a small box, fit a small bass driver, and the result is inevitable. For this reason, I would hunt down a good used pair of Monitor Audio Bronze 2 or BX2 (previous model). Any sub you can buy for under £250 new ain't going to do much better than these for bass. In order to make an appreciable improvement then your sub budget would need to be £400+.
£400 on a 5.1 AV surround system just won't compete for music. For a start, you're spreading the speaker money really thin. Let's say for £200 you can make two speakers or 6, then it stands to reason that each of the 6 speakers gets a much thinner slice of the money cake. All else being equal, money equates to sound quality. Less money equals less sound quality.