I'm not going to go through what I'd buy for myself cos we have different lifestyles and probably different tastes. But talking generally, there's no way I'd try and replace my whole wardrobe at once unless I'd literally lost everything but the clothes on my back. Because I'd make a lot of panic, 'that'll do' buys and probably end up with a wardrobe that wasn't much better than the one I threw out.
Also Trinny and Susannah's show was early 2000s which was arguably the golden era for the UK high street - prices were low, quality was still pretty good and 2K could get you a long way. It's nothing like that now and, while I think there are still bargains to be had, it can take a lot of digging plus a big slice of luck to find them.
So I'd treat it as a medium term project, meaning I'd expect it to take at the very least 2 years and preferably 5.
I'd start with 3 lists:
First, the cheap and cheerful - stuff I'd be happy to buy in supermarkets, H&M, Uniqlo etc and that I'd accept would need replacing fairly regularly. I'd probably include jeans here. (I know some people will disagree but I can't honestly say I've noticed a huge difference in quality between Sainsbury's and high end.) Also a few fashiony things that would by nature have a limited shelf life. (Yes I know, bad, bad me.)
Second list: things I'd pay more for because I've learned the hard way that it's a false economy to try and buy cheaply. I'd include shoes here and also a bad weather, waterproof or at least water resistant coat but you might have other priorities.
The third list would be things I'd happily buy second hand. This would be quite a long list because it's how I buy most of my clothes but it's a random way of shopping and it might take ages to find what I want so I might well have to buy a cheaper version as an interim. For example, I'd include a wool, winter coat here.
My thoughts on Vinted: Yes it can be brilliant. It's also full of crap that you can't return and it's highly addictive. With a limited budget I'd use it, but sparingly.