It sounds like an assortment of dirty, corroded or oily electrical connections to me.  

I'd start with cleaning all the electrical connections, especially the engine ground strap, the the body and the engine where the strap is fixed, and the plug connecting to the alternator to ensure that once it does start, the newly replaced battery will charge when the engine is running.

Why did you replace the starter? How did you decide that there was an issue with the old starter?

A simple test lamp, even a bulb with wires held on poorly with tape, can determine if the circuits are completed to trigger the starter: ignition switch triggers the relay, the relay triggers the starter.

The later Minis with pre-engaged starters like your car should have used a starter relay which in turn sends a trigger signal to the starter solenoid.  You can test the solenoid in a couple of ways.  I'll suggest one test but before posting more, it would help if you can post pictures of your engine bay and the starter motor.

1. Put the car in neutral and set the parking brake.
   Open the bonnet and look at the terminals on the starter motor. You should find a small spade terminal with a brown/red wire on it.  Remove that wire.
   Make a short jumper wire with the insulation stripped from both ends.
   Connect one end of the jumper to the spade terminal where you removed the brown/red wire.
   TAP the other end of the wire to the threaded solenoid terminal with all the brown wires on it.
   
   If the starter motor does not turn over when you tap the wire to the solenoid terminal, the problem is with the starter motor.
   If the starter motor does turn over when you do this test, you have a problem with either the starter relay or the ignition switch (probably the relay).

Sorry for being unclear. I am still trying to learn about engines and vehicles  so I will try to better explain myself. When I turn my key in the ignition the engine does not turn over at all there is only a single click comin from the relay switch. After replacing the starter with the integral solenoid I am still getting the single click. I will check into the solutions you mentioned though. Thank you

I had a similar problem (sometimes the car would start, other times just a "click") which was the cable between the solenoid and starter.  It had some corrosion which by looking at it appeared to be fine.  After cleaning it up not a problem since.

Post your location (if you're comfortable doing such a thing) as there may be a fellow enthusiast nearby that can lend a hand.

So, if you turned the ignition on, let the car roll backwards down the driveway in reverse and released the clutch, the car would start? I mean, it sounds as though the starter does not work when you turn the key, but if it did work, would the engine even start?

Or, is the first problem an inability to describe the problem?

If the starter is new, then it should be able to turn the engine over. If the starter is for a 1990 mini, it should have an integral solenoid, yet you say the solenoid has been changed as well. This could be the case, but a well described problem would point out that the 1990 mini in question is using an older starter system with a remote solenoid.

It sounds like the original 1990 wiring harness, as you mention a relay that then triggers the starter solenoid on the starter, which then powers the starter motor. I guess you want to see that the relay gets triggered by the ignition switch, that it then triggers the solenoid, and that the starter is then powered. If all of those things are going on, then it sounds like a weak engine ground is your issue.

If the car is actually a much newer model than a 1990, as sometimes happens when unscrupulous people change VIN numbers to sneak cars into markets where they would not normally be allowed, then it is even harder to troubleshoot via the Internet, and again that takes us back to discussing the value of describing the problem.

A simple test lamp, a workshop manual with the page opened to the wiring diagram, and a couple of hand tools should have your starter motor functioning in a few minutes. A bad engine ground strap would easily cause issues like this.

Hello everyone,

     I am pretty new to the mini community but I recently purchased a 1990 Austin mini. I drove it for a few months and it has been great. About a 3 weeks ago I was warming the engine up for about 10 minutes and stalled trying to reverse. Went to turn the key back over and all I hear is a single click. My battery, starter and solenoid have been replaced and I am still having the same issue. The click sounds to be coming from the relay switch. Any advice?