Car cut out like no ignition, now no power to starter circuit. Battery new & hot

Is the car now running? If it is, then what you need to do is get a hold of the car schematic and check out all the components in between your power source and the items not working. Use a voltmeter or 12 volt car tester lite to track down the issue.

Could be issue with ignition switch or security system? If the starter motor won't crank, check voltage circuits at the starter. Usually two voltage circuits, one is battery voltage, hot all the time. The other voltage comes from crank position of ignition switch. That circuit could be routed through the neutral switch or clutch switch, depending on your tranny.

Any testing at starter make sure tranny is in park or neutral and the parking brake is set.

Before you do any thing check for loose or bad connections. Also check you ground connection to engine. Check Fuses, relay and for a Fusible Link open circuit. Now let me explain what a Fusible Link does. They are short lengths of small gauge wire installed between the Battery and the Ignition Switch, or the Battery and the Fuse Panel. They'll melt and open the circuit before damage occurs in the rest of the circuit. I hope this was helpful you.

There is no starter relay shown in a wiring diagram. The only thing between the ignition switch and the starter solenoid is the park/neutral switch (or the clutch pedal position switch on manuals).

Do the warning lights and gauges come on with key on? That would verify power is into the switch and being sent out. The small purple wire on the starter solenoid should have power on it when the key is held in start position. If it does, recheck your starter and battery connections. When the purple wire energizes the solenoid, the starter motor should crank the engine.
If no power on the purple wire in start, check the neutral safety switch or the clutch switch if a manual. The start wire out of the ignition switch is a yellow wire, and the wire from either safety switch to the starter solenoid is a purple wire.
(The power to the ignition switch is from the 50 amp Ignition fuse in the under hood fuse block.) See? The positive battery cable is connected to the starter, but not to the starter motor. When the key is in start, the hot purple wire energizes the solenoid and the battery cable is connected to the starter motor.

Let's see if I can explain it. Pretty simple start circuit, but starting issues are very common to Hondas and Toyotas of the '80's and early'90's. Lots of guys used to rewire their start circuits with new wiring and relays from ignition switch to the starter solenoid just to stop those no starts.
From the ignition switch is the black/white start wire, it is hot only in start, goes direct to the "starter cut relay"- a 4 pole starter relay, the black/ white wire (black with a white stripe) is both the power feed and the coil power to the relay. Before the wire enters the relay it is spliced. one wire goes to power feed at relay, other wire goes to coil side of relay. Here's where the clutch switch comes in: the clutch switch acts as the ground for the relay's coil side. Okay, you turn key to start, power goes to the relay on both the power side and the coil side. When clutch is depressed, it grounds the coil side of relay, this energizes the relay, so power goes out of relay on the power side, as a black/red wire...and goes direct to the starter solenoid. Somewhere before it reaches the solenoid, it turns back into a black/white wire. But nothing else is in a wiring diagram for your start circuit. Buy a can of CRC electronic cleaner. Spray all connectors in the start circuit. Pull off the solenoid wire at starter-that connector. Also the starter relay , pull it out and clean the relay terminals where it plugs in. Get all clean connections in the circuit. If no help, swap in a different relay. If still no help, get a voltmeter (or a test light), when the starter doesn't want to work, check for power at the starter solenoid with key held in start, and clutch depressed. If power there, the solenoid is the problem. If no power, work back to the relay. Pull relay out. With key held in start, two terminals for the relay should have power-the coil plus side, and the power feed from the ignition switch. If both have power, may be the clutch switch not grounding the relays coil side. If no power to relay, check the start wire out of the ignition switch.
Need a wiring diagram? This site is awesome-free wir. diagrams and TSB's for every make and model:

I am assuming you are talking about the circuit that powers the key switch. I'm showing a fuseable link to the starter relay instead of a fuse.
If it is the circuit that powers the switch, I would disconnect the switch and see if the fuse blows as each circuit is powered one at a time using a jumper wire. You can get 30amp circuit breakers that plug into the fuse slot to use while you are testing circuits.
You may need a repair book or shop manual to find out what is powered by that fuse since it is most everything on the car.

Did you check both the positive and negative battery cables? The two fuses on the right are voltage for ignition switch. Use a test light, see if the fuse circuits are hot? Should be hot all the time. There are fusible links involved, also. The battery junction box is under the hood.

You definitely have a starting system problem. You have two circuits - Battery to starter circuit & the Starter Control Circuit (Ignition switch, Transaxel Range Switch (if Automatic Trans), Starter Relay, Starter Clutch Pedal Position Switch (if manual Trans) Fuse & Starter Solenoid). Based on your description, the problem is in the Control Circuit.

Here's an illustration:

Check fuse # 28 (15 Amp) with a test light. (Located in the Power Distribution Center (PDC) on the left side of the engine compartment) It should have power only when the ignition switch is turned to the START position. If there is no power there, then you probably have a bad ignition switch. The only other fuse involved would be fuse #8 (40 Amp) which is also in the PDC. If this fuse is blown a whole lot of stuff is not going to work - not just your starter. The starter motor relay (Also in the PDC) is controlled by the PowertrainControl Module. The wiring diagrams show it as a 5 pin relay whereonly 4 of the pins are used. If you check for power and ground at the connector pins for the relay, there should be one pin that is hot all the time. This is the power from fuse #8. a second pin should be hot with the key in the RUN and START positions. This would be the circuit running through fuse# 28. One pin should have continuity to ground (your test light should light up if you hook it to the battery positive terminal and probe this terminal.) This is the circuit that goes down to the starter. The 4th pin will have no continuity to anything at all until the ignition is placed in the START position, when it will have continuity to ground. This is the relay control circuit from the PCM.
If you dont have power at the relay connector as described, then you must repair the circuits between the fuses and the relay. If you don't have the grounds as describedfor the starter circuit, you ether have an open circuit going to the starter or a bad starter. If you are not getting ground on the control circuit from the PCM, it could be several other things causing your no-crank problem. It could be an open circuit between the starter relay and the PCM or it could be that the PCM is not allowing an engine start for one reason or another - The PCM will have to be scanned to find out what the problem is.

There is a circuit staying open that is draining the battery. Chances are it is the ignition switch or the starter relay. Most likely it's the ignition switch. If it stays open the starter circuit continues to draw power even though the car is off. It's under the dashboard above the gas pedal. Someone should be able to test it with a circuit tester to see if it's the problem. Good luck!