Since I didn't see a post regarding Centurion UK Criteria transferred in here, the last I heard of are as follows (though these are probably quite old data points). Amex does not publicise the criteria and of course they are not hard and fast rules and Amex can pick and choose whomever they like:

- It is invitation only. You can always call up and state your interest but that doesn't guarantee anything.
- There are no defined number of issuances each year. Amex send invites when they want to. Presumably they control numbers. Like any business they need to be profitable and viable and ensure quality of service.
- To qualify you need to have an Amex history. Generally it was considered that The Platinum Card (charge card) held for at least one year with no issues was a requirement, but it seems from posts above that invites have gone out where a holder had only e.g. a corporate card (?). In any event, if you are going to try and get one, I would personally suggest getting The Platinum Card ASAP (before cost goes up shortly!) and start using..... it is worth using the services, Platinum Travel Service (they get commissions on bookings) and Concierge and the like.
- There used to be an income requirement of c.£300k/year but that may have increased and I'm sure is flexible. That said if you spend the levels required you probably earn more than that anyway.
- You will likely have a credit check undertaken and may need to even show proof of income and proof of liquidity (for higher spend thresholds). You may get the invite and subsequently have to undergo a credit check after you have accepted, which seem odd, but its prudent to ensure you aren't spending money that you really don't have.
- Annual spend hurdle at one point was more than £125k p.a. but it wouldn't surprise me if that is now more like £250k+ (the old Medallion threshold). That said if Medallion was considered the upper tier of Centurion, I'm not sure how many Cent holders would qualify today.....
- It is generally believed that use of the card on general day to day purchases is a plus. i.e. spend on travel, luxury goods, starbucks coffees, petrol pumps, is what they like to see. Spending £100k in one month on paper clips not so much. High-end personal spend which they can market to their clients.
- While officially its "unlimited", cards all have some limit. For example if I have more than £100k outstanding in any month, they ask for more financial information from me. I politely decline and pay my card down. With the Plat card it works on some algorithm that your "limit" is 150% of your average highest rolling 3 month spend or something like that. It can take some time to build this up. With the Cent you don't have that so much as they set some higher thresholds I believe. At the end of the day they don't want a Cent declined.....
- The cost today is £3k joining and £2.2k p.a. thereafter. The cost increases sporadically every 5 years or so.

As noted many times on FT, there are always exceptions. Someone who never held an Amex card. Someone who spent £3 on a Green Card last year. I'm sure there is more to these stories. But if you want to have a real go for it, I suggest targeting the above for your best chance of getting an invite.