Is lithium battery charging speed proportional to the amps provided from the power source?
In theory in the US you get 20 Amps from a wall socket.


some sockets in the US are 20A but many are only 15A. So a portable appliance vendor who limited their product to 20A sockets would be limiting their market quite severely.

My understanding is that all sockets in Australia are at least 10A. When you look at other ~230V first world countries you mostly see sockets ranging from about 10A to 16A. So someone designing an appliance for 10A@240V (~9.6A@230V) can sell it across most 230V first world countries.


  An interesting fact I came across recently - electric kettles in the US run at double the time for other countries - due to the US adopting a 110V power system. (And electrical kettle boiling duration being proportional wattage available). (1500W in US and 3000W in other countries). 


Kettles are unusual among domestic appliances in being both high power and portable. They tend to use most of the power available from an ordinary domestic socket. So weedy domestic sockets mean weedy kettles.


  I'm wondering how this applies to charging batteries.


Batteries in portable appliances are another story. Typically the maximum safe charging current for a Lithium ION battery in amps is typically similar to the capacity in AH (in battery charging jargon this is referred to as a charge rate of "1C"). 

So lets consider a phone battery. Lets say you have a phone with a relatively big battery. 3AH. Maximum charge current for such a battery would be about 3A. In practice the phone vendor may limit it to 2A due to thermal considerations and concerns about how much current the connector can handle.

The charging circuit in the phone is typically supplied with 5V to allow some overhead over the battery voltage for charge regulation. So say 5V 2A is 10W.

We quickly see that compared to what a typical domestic socket can supply this is bugger all. The domestic socket is not the limiting factor here.

A laptop battery can be bigger. Say 6AH with three cells in series. The laptop power brick usually supplies about 20V but given the larger power in a laptop the charging circuit will almost certainly be a switched-mode converter. Battery voltage while charging will be something like 12V.

6A at 12V is about 72W. Add a bit for inefficiency in the charge controls and for actually running the laptop and a big laptop might pull 120W or so from its power brick. The power brick in turn might pull 150W or so from the wall.

A bit bigger than the phone but still small compared to what a domestic socket can supply.


  That leaves me with a question - do batteries take twice as long to charge in Australia as the US? 


No.

Firstly the battery charge current has nothing to do with the mains current. There will be a power supply in the charger which can trade off voltage for current.

Secondly as mentioned above the domestic socket is not normally the limiting factor for charging portable appliances.

An exception may be electric car batteries. Charging an electric car off a normal domestic socket is going to take longer in countries where normal domestic sockets are weedier.

Having said that a lot of electric car charging is not done from ordinary domestic sockets.


  If I move from Australia to the US will my phone charge faster?


No.

If anything most universal voltage supplies will work better on 240V input but this will only make a difference in cases where the power supply is crappy enough to be the limiting factor and even then it will be marginal.