Will this DC UPS with off the shelf components for home network and laptop work?
A solar charge controller isn't ideal for managing the battery.
A basic controller like that works by alternately connecting the panels directly to the battery, and disconnecting them. This is only OK because the solar panels are a current source. The solar controller isn't capable of floating the battery, so when the bulk charge is finished it just squirts in a bit of high current every now and then.
The buck converter you show is actually capable of current limiting, so it won't blow up.
But the battery will take a beating, being alternately charged and discharged every second or two, as it supplies the loads, then the voltage drops, and the charge controller connects the 20 A current source, only to find the voltage shooting up again.
The way you usually make such a UPS is like this:
15 V DC source -&gt; diode -&gt; load  
15 V DC source -&gt; linear charger -&gt; battery  
       battery -&gt; diode -&gt; load

This way when the mains power is on, there is no other current flowing past or through the battery, and the charger can do the right thing. The loads are powered directly from the DC supply, via the first diode. When the power turns off, then the second diode will supply the load from the battery. Diodes should be Schottkey to reduce the voltage drop.
You wouldn't use a 24 V supply, that would waste too much energy in the (linear) battery charger, and it would require your laptop supply to be a buck-boost to cope with +24 V when the power is on.
The rest looks sort-of OK. Be sure to put at least one fuse between the battery + terminal and everything else, so you don't start a fire if something shorts out. And watch out with these cheap and cheerful SMPS boards, I have found them to overheat at well below their claimed rating. I recommend getting them rated at twice as much current as you think you'll need. One clue is simply physical size - if it's a lot smaller and lighter than your current laptop power supply, then it probably won't cope.