I have a couple of turntables here and have been doing the vinyl thing for the better part of 30 years so I've seen the ebb and flow of popularity in records. Right now vinyl is very much 'on trend'.
Back in the 80's when I first started my Hi-Fi journey vinyl was the medium of choice for quality, and the recommendation was to spend the biggest chunk of a budget on the TT/arm/cartridge combo and then progressively smaller amounts on the amp and speakers. There's still some sense still in that approach; putting really good speakers on the end of a mediocre TT and amp combo would just highlight all the deficiencies further up the chain. Having said that, manufacturing quality has improved and the cost of manufacturing has fallen to the point where £200 on a single product today buys better performance than was ever possible with an inflation adjusted equivalent amount 30 years ago.
The entry-level turntable of choice back in the mid 80s was a Dual CS505 mkII. It cost around £110 at the time. In today's money that's the equivalent of approx £350. Today you can walk in to a Hi-Fi dealer's premises and buy a Pro-Ject Elemental for £159 that would give the Dual a damned good run for its money. In the mid 80s that would have been like spending about £55, so roughly half the price. That's how much things have got cheaper in real terms. It's a similar story with amps and speakers.
Back to today, if you want the best bang for your budget then I'd seriously consider mixing new and second-hand.
It used to be possible to used turntables really quite cheap, and some really good products too such as Rega Planar 2 and Planar 3. These were the 'go-to' second-hand turntables of choice. The hipsters have changed all that. Planar 2's and P2's that used to be available for £60-£80 are now fetching anything up to £200+ and typically sell at £140-£170. Used Planar 3's fetch £300.If you're looking for value in the used market then you should focus on Pro-ject turntables, and personally I'd search for Project Debut in original or mkII or mkIII form. As the 'debut' name suggests, this used to be the entry-level product in their range but now there are two lower priced products: the Essential and the more bare bones Elemental. Debuts have been around long enough for there to be decent amounts of used gear available. The used prices are still sensible (£70-£150), but if you prefer to buy new then a Debut SE3 can be had for £265 including cartridge.
For an amp you need something with a 'phono' input for a turntable. That's important because the sound coming out of the cartridge is a very low signal level and it has almost no bass. The phono input has a special circuit that boosts the power far more than an ordinary line input (CD, AUX etc) does and it also rebalances the sound. For new amps I'd look at the Marantz PM5005 (£170), Yamaha AS201 (£149). Used amps in the price range from £50-£120 such as the NAD3020 and C320BEE, Rotel RA820, Marantz PM7003, ARCAM Alpha will give you £200-£500 amp performance.
If sticking with all new then the remaining budget for speakers is £65-£85. There are speakers available in that price range but IMO most won't do the system justice. It's just too little money which means too many compromises. The only ones I'd consider would be Tannoy Mercury V1 just because they're on clearance at the moment. Personally though I think Wharfedale Diamond 220 make a better long term choice, but they're £180/pr. Used speakers in the £50-£100 price range opens up all sorts of interesting options including decent floorstanders from KEF, Monitor Audio, Mission and others.