I did read in the instructions that if you have any jumpers to not use them with the Nest- and this would seem to confirm that as well -
https://support.google.com/googlenest/answer/9250973?hl=enSo installation should be as simple as Red to RH (or just R), Yellow to Y, Green to G, and White to W
I believe power comes from red, are the batteries in your original thermostat just a backup for your settings possibly?
It is fully programmable with the Nest app and you can do away with the turning on and off when you leave the house and just run it more like a regular programmable thermostat. I will be doing the same thing, as my wife is working from home all the time for now whereas I still go in to the office a couple of times per week.
Not trying to sell you on one brand over another.