will the circuit attached work to charge the 4.8V li-ion batteries in series?
No, very definitely not!
It would be very good at causing them to burst or explode....
The circuit shows a constant-current charger, which with a low enough current setting can be used for NiCd or NiMH cells, with the batteries being removed after an appropriate time. That type of cell can stand a
small current for very long periods, after reaching full charge.
Charging lithium cells is more like blowing up balloons - if the current, even very small, continues after they are full, things go bang! (And burst lithium cells quite often catch fire spontaneously).
Also, as has been mentioned, if you are charging two or more cells in series, "Balance charging" is essential otherwise the burst cell situation is still possible even with the correct overall voltage.
Lithium cells must be used in combination with undervoltage/overvoltage/overcurrent protection modules, often listed as "BMS" modules.
And, for a battery pack with two or more cells, that BMS module must also incorporate cell balance.
The BMS is not a charge controller, only protection - you still need a charger with the correct voltage and current limits for the battery pack being charged.
Failing to properly charge and use lithium cells can be lethal - the energy content of fully charged cells is approximately equivalent to a quarter their weight of TNT... They may not release it quite as fast, but they can do just as much damage if misused.