There are a few options for affected customers (ie those booked to fly BA between 20 December and 4 January), although these will not necessarily solve all issues.
1) Right to cancel with cancellation penalties waived.
2) Right to change dates within 12 months with change fee waived.
3) If your ticket is a flexible fare (eg a Oneworld Explorer) you may be able to reroute, switch airlines or change dates in accordance with fare rules.
4) When flights are cancelled you can be rebooked on other airlines.
So far BA has not yet cancelled flights but instead has blocked every flight from being sold (to cap the number of people they need to deal with). People with affected flights who have the means (ie lots of $$) to book refundable tickets on other airlines are doing so because there may be no options available by the time BA gets around to cancelling their flight. On many routes which British Airways flies the flights operated by other airlines are now either sold out or have only very expensive fares available for purchase.
The situation is fluid and will change from day to day. So it is important to keep informed (refer to news and advice on BA.com and sites such as Flyer Talk) and also to be in close contact with your travel agent if your ticket was bought through one.