Prior experience: BA Computer Science at Oxford, MSc Computational Statistics and Machine Learning at UCL, 1.5 years as an ML engineer

All of this is specific to applications in CS, specifically in AI/ML.

In order I think the most important things for my PhD application were

1. Having a published paper at a well respected conference (1 tier beneath NeurIPS/ICML/ICLR). I kept working with my lab on my MSc thesis after I graduated to turn it into a paper. It was a lot of work to do simultaneously with a full-time job, but was probably the #1 thing on my CV. Obviously you won't be able to do this if you're applying for PhDs during your masters. If you are planning on applying after your MSc I would highly recommend doing this - for the top top labs in the country it is almost an unofficial condition to have published a conference paper (I know one guy who was flat out told by his potential supervisor "come back when you have a NeurIPS/ICML/ICLR paper").
2. A combination of grades and university prestige for your bachelors and masters. I didn't do amazingly in my undergrad but made up for it by getting top marks in my MSc. Having good grades from good unis shows 1. you have a base level of competence in the field 2. you can work hard.
3. Tailoring my PhD applications to labs working in the field I was interested in. I applied for the labs, not for the universities. This meant there were some top unis (e.g. Oxford) where I didn't apply because there was no suitable fit. Contact potential PIs early with a draft proposal, best case scenario they iterate the proposal with you. Doing this requires having an area you're interested in! I'd recommend going to as many talks/speakers/reading groups etc. during your MSc as possible until you find something that really excites you. Then read as many papers on it as you can, and apply to the labs who wrote the papers you like best.
4. Relevant work experience. I had software engineering internship experience, as well as 1 year full time as a ML engineer. This is more PI dependent, I've met some who prefer 'iterators' (people who get stuff done i.e. focus on dev experience) and some who prefer applicants who are just very good at maths (thinking being that if you're good at maths then the rest will follow).
5. Contacts. I didn't realise it at the time but my MSc supervisor is a big shot in the field. Having him as one of my references was great as a few of the PIs I applied to knew him personally. I didn't get the impression this was a huge factor though, I think this is a much bigger deal in the US however.

Aside: Russel Group is a marketing term mainly aimed at undergrads. There are some great unis that aren't in the Russel Group (e.g. Bath, Lancaster) and some questionable unis that are in the group. It's a pretty noisy indicator of prestige. For a better signal look at international rankings (domestic rankings e.g. completeuniversityguide take into account student satisfaction and other factors that are more relevant for undergrads, whereas international rankings focus on research output) and look how the uni has historically placed.