I don't know how a British timeshare works, but I know a little about US timeshares and condotels. This response is based on that knowledge. It may be different in London so do your own due diligence, this is for informational purposes only.
It sounds like condotel/time share hotel.
If you buy outright (condotel), I'm assuming that you can put the room into the hotel room pool when you are not using it and get a % of income. In that case, you have to see who controls different aspects of the hotel like the front desk, parking and other income sources like vending machines. Does the management company own these resources or do the owners (aka you)?
If the owners own it, then you have control of the income and how things are run. Of particular interest to you as an owner is the front desk. If the management company owns the front desk, they will probably give preferential treatment to the other rooms and fill the owners rooms last. If you don't have a say in this process, then you could get screwed in off peak times.
Because it looks like they offer both condotel and timeshare, I'm guessing the management company owns everything.
If you go the timeshare route, just watch for the annual fees and if the rent, after expenses, will make you money.
Also check to see if you can exchange your quarterly points for a stay at other timeshares around the world. That might make it worthwhile.
A timeshare can be a cheap way to get in, but keep in mind that if your other timeshare owners don't pay the annual fees and property tax, the other 12 of you will be on the hook for those expenses. They can foreclose on the timeshare.
In addition, many timeshares in the US have a first right of refusal clause that states that the hotel gets first crack at buying your share back...at the price you bought it for. That doesn't allow you to make a profit.
Anyway, just a few thoughts. Again, do your own homework, but hopefully I pointed out a few pitfalls.