Mini just wont start!

N18 engine spells IBS system to me. The battery has to be re assigned to the computer or something. from what I have been told.
If no IBS system is in place I would take Voltage meter and check the
connection for power while someone presses the start button and presses on the brake at the same time you should have 11v. or something like. If not the connection is bad. if so then the new starter which is not uncommon is bad or not working with your setup.

When dealing with any of these issues I have found that yes. One can take out the other when you introduce a new one of the three into the mix (Alternator,starter,battery).
Your new battery can go when you put a new starter on it. Your alternator can go when you put a new battery on it. Your battery can go when you put a new alternator on it. And any mix of any of those. Can cause one or the other or all three to go out. whether they are new or not.
It's a touchy area to work on. Working with a local parts store makes it a bit easier, at that point you can return the alternator or starter or battery in a matter of minutes.
If you know you vin code on your car and know how people that order batteries for the store front stock because they specialize in selling car batteries. Then you would be able to find a mini cooper battery for your car. There are dozens of different types of batteries for mini cooper. there are a whole assortment just for the R56. It's comes down to a model specific battery more than just a R56 battery.
Now even though it can turn into a wildfire of sorts when dealing with a modern electronic area. What's said is said when you decide to introduce those idea's. Meaning it can be nothing other than.
those three items. provided you did not mess with any wiring. That is where it stops. Because it is a modern car. To question other things aside of that connection to the starter, that IBS 50 amp fuse or something like that, And the proper continuity of the Alternator, becomes a very very slim chance of occurring.
Here is a chart at this point I have to agree with Mike1967

These are where all your grounds on your car are located at. It MAY be something going on with all that. It would be more probably. But the starter is best bought genuine mini cooper and returned through the mail if it does not work off the bat. That happened to me once. If you get a off brand do that with your local parts store. Battery is the same way. Only a little harder seeing how it is a lot harder to find one model even vin specific. Dealership item is what it used to be back in 2010. Now in days it's more widely available. I don't imagine you have a Wal Mart there? they could probalby help that. But I went to a actual Battery store. Where they sell all types of batteries car batteries being a lot of there stock also.

It should be hanging off the positive battery terminal. I think it is a 50 It could be a 150. I do not have the IBS system. But I have looked for it in my car. You have to take off the plastic shroud (cowl) over the battery to access it. But I do know they came out with the IBS system like a year after my car 2009 or something.
And as I was typing to other threads. Sometimes connections have to be removed and redone.
I just responded to another thread. From the overflow of others having start issues.
This was my re fined response at the tail end of this thread I just responded to.
It is important in the cold and windy type weather. And it is surprising as I actually never thought about it.
Until my other car went to a mechanic for it. That was literally all he did to get that car to start.
I paid him 200 U.S. after it was said and done. Either way it is good practice to do all available ground connections in this way.