Wow! Sure looks cleaner that the original. All of you ROCK!

Thanks for taking the time to do that. The relays are 5VDC, not 3 VDC as the schematic shows. Sorry about that. My wife is already aggravated with me for spending so much money on all my projects, so I have to work with the parts I have. So I have some questions, if it's okay. Hopefully they haven't already been answered and I'm just not understanding, but here goes.
1. How is the battery circuit integrated? I see that the output / input has the BAT symbol on it, but I'm not sure how that works.
2. What size / type of mosfet, and do I need one for each relay (I assume I do)?
3. My wall wart outputs 12.6 volts steadily when checked with a multimeter, so is it appropriate to go from the wall wart directly into the buck converter to level it out a bit more?
4. I was under the impression that I could tie the output of the other buck converter in with the 12v input from the wall wart. If not, what's the best way to accomplish this?
And to answer questions:
The relays you used in the schematic are listed as 3V instead of 5V, so if these need to be used a separate 3V supply needs to be created. 12V relays would be better though.
I have an abundance of 5VDC relays, so if at all possible, I'd like to stick with these.
What is the voltage of the wall wart you are using. I know you wrote 12V, but it could be a non stabilized supply that has a higher output voltage when not loaded, or a switched power supply.
Steady at 12.6v according to a multimeter
Then there is the issue of the power switch. What does it need to control? Just the Arduino Nano system, or also the charging of the battery?
The power switch should turn everything other than the charging system on or off. If I plug the wall wart in, the batteries should charge. I purchased two boards to accomplish this, wasn't sure which one to use so I got them both.
https://www.amazon.com/dp/B0C6DNMKS7?ref=ppx_yo2ov_dt_b_fed_asin_titlehttps://www.amazon.com/dp/B08M36F6XD?ref=ppx_yo2ov_dt_b_fed_asin_titleAnd the batteries I purchased are as follows.
https://www.amazon.com/dp/B0CLY68D5L?ref=ppx_yo2ov_dt_b_fed_asin_title&th=1For the battery backup you did not put in a charging option, and connecting the output of a buck converter directly to the 12V input is not good practice. Diodes will be needed there.
I thought the charging option would come from the 12v wall wart. What type of diodes should I use for this, and how should I make the connections? During storms my power almost always goes out. I have a 13Kw generator that kicks on automatically but there is lag time from mains to generator. A little background on why I made this unit. My garage is detached from the house and sits to the south of the main entrance way, which means that if I want to know if the garage door is open, I have to step outside and look. So I came up with a device that uses a reed switch to detect the door being open or closed. If the LED is red, the door is open, if green then it's closed. The problem is when the unit loses power, it switches to red even though the door may be closed. I have no way of knowing unless I either look outside at the door, or press the open / close button to trigger a change in the code. You all probably didn't really need to know all that, but I wanted to explain it so you understood what I was doing. Again, I just want to thank you all from the bottom of my heart for helping me, and yes, I'm learning a lot by your questions and examples. Enough to know that I completely botched the schematic.
