Firstly, whether or not water breathing is magical enough to effectively 'make' oxygen where there is none is debateable. Other answers have tried to deal with that fact, mostly concluding that it's up to you. The premise of this answer is that water breathing only allows you to breathe oxygen that is already in the water. It attempts to calculate how long a human could survive on just the oxygen dissolved in the average waterskin.
DISCLAIMER: This is just my attempt at a calculation; if you spot any errors, and there probably will be at least one, please do tell me.
The PHB says that a waterskin holds 4 pints of liquid (probably fresh water). 4 pints is about 2.3L. Assuming that the temperature is about 20°C (just above room temperature), fresh water can hold a maximum of 10mg/L dissolved oxygen (using this data about sea and fresh water).
Our waterskin therefore holds, assuming the 'best-case scenario', about 23mg dissolved oxygen. 23mg is 0.023g
An adult needs about 840g oxygen per day (according to NASA). That's 35g per hour, 0.58g per minute, or 0.01g per second. Based on these calculations, an adult could breathe in sufficient oxygen for 2.3 seconds from the waterskin before running out.
Using the suffocation rules in the PHB (pg. 183), and assuming that the character has a CON of 14, they could hold their breath for 3 minutes, then could last 12 seconds (two rounds) before dropping to 0hp.
Using this method, a character could expect to stay conscious for 3 minutes and 14.3 seconds before dropping. After that, by the rules, the character could become stable through death saving throws.
In reality, the results would be more serious (see this source). After 15 minutes, recovery becomes virtually impossible as too many brain cells have died.
Your character is therefore likely to survive 18 minutes and 14.3 seconds before being so deprived of oxygen that they might as well be dead. Not a particularly effective strategy all things considered...