Bleaches work by oxidising substances. They remove stains and sanitise by breaking bonds within a molecule. This produces smaller fragments that do not absorb light in the visible region of light (with respect to human sight), making the broken fragments colourless.
What is oxygen bleach?
Oxygen bleach is sodium percarbonate (Na2H3CO6). When sodium percarbonate is combined with water, it forms sodium carbonate (Na2CO3) and hydrogen peroxide (H2O2), hydrogen peroxide is the sanitising agent. Sodium percarbonate is one of the active ingredients in Vanish (previously known as Napisan).
What is chlorine bleach?
Chlorine bleach is sodium hypochlorite (NaClO) diluted in water to produce a certain concentration eg 1%, 4.5% etc. It is standard household bleach eg. White King or the generic supermarket brand.
Chlorine bleach is unstable and reactive, which is why it has a shelf life, eventually turning into saltwater.
Sodium hypochlorite is effective against bacteria, viruses and fungi, sanitising the same way as chlorine does. 
Chlorine kills pathogens such as bacteria and viruses by breaking the chemical bonds within the organism. When enzymes come in contact with chlorine, the hydrogens in the molecule are replaced by chlorine. This causes the entire molecule to change shape or collapse. When enzymes do not function properly, the organism will die.
The most effective, easy and inexpensive sanitisation options are chlorine bleach and 90/95°C sanitise options. Chlorine bleach is bacteriocidal (ie. kills bacteria) at low concentrations, and both oxidise ammonia.
Oxygen bleach it not a viable alternative to chlorine bleach or the 90/95°C cycle sanitise options.
Oxygen bleach is not an effective sanitiser for cloth nappies
Approximately 30% of the solution would need to be Vanish, so for every 1L of water, 300gm of Vanish is required. Sodium percarbonate is soluble at 120-150g per 1L of water . This means that over half the Vanish required to be able to sanitise would not actually dissolve.
If solubility was not an inhibiting factor, to sanitise using Vanish in a standard wash cycle:
10-20kg is needed to produce enough hydrogen peroxide, at a cost of $5-$8/kg. The approximate cost for a standard* front loader wash cycle: $60 The approximate cost for a standard* toploader wash cycle: $90
In comparison to using chlorine bleach to sanitise in a standard wash cycle:
80-120ml of chlorine bleach, at a cost of $2- $4/L The approximate cost for a standard* Front Loader wash cycle: $0.45 The approximate cost for a standard* Top Loader wash cycle: $0.80
Oxygen bleach is not a viable alternative to chlorine bleach or the 90/95°C cycle sanitise options.
History of Napisan
In the 1970s and 80s, the active ingredient in Napisan was potassium monopersulfate (KHSO5) which oxidises sodium chloride to sodium hypochlorite (chlorine bleach). The formulation has since changed, with sodium hypochlorite replaced with sodium percarbonate (oxygen bleach) .
*Wash cycles vary from 60-75L of water used for a 7-8kg Front Loader, and 80-110L for the equivalent 7-8kg Top Loader. Wash cycles contain at least a wash, rinse, spin phase, with sanitising occurring in the wash phase. Cost estimates are approximating the wash phase to account for ⅓ of the total water usage per cycle.
 Benckiser, R. Chemistry in Your Cupboard: Vanish- Learn Chemistry
 H. Gaya et al, J. Hyg., Camb. 1979, 82,463
 K.T. Miyasaki, R.J. Genco, M.E. Wilson (1986). Antimicrobial Properties of Hydrogen Peroxide and Sodium Bicarbonate Individually and in Combination Against Selected Oral, Gram-negative, Facultative Bacteria
 M.G.C. Baldry (1983). The bactericidal, fungicidal and sporicidal properties of hydrogen peroxide and peracetic acid
 Solvay Sodium Percarbonate MSDS