Because of the scale of the disruption resulting from an IT system outage, most passengers should expect to make their own arrangements to mitigate the disruption as BA will not have resources to assist all passengers.

Your travel plans are probably in tatters, you should give BA a chance to fix things for you.

Firstly do not cancel or refund or credit to a voucher your ticket if this option exists as you effectively voluntarily cancel the contract for transportation with BA by doing so and hence end any right to claim for replacement transport, duty of care, or anything else.
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If BA.com is working, at least give it a go and see if Manage my Booking is giving you any sensible options. You don't have to take the option(s) presented if they do not meet your needs and you might be able to find better options yourself. However, not that by default BA will be only offering rebooking onto BA and any Joint-Ventre carriers and possible OneWorld partners. If there are no sensible options, document it (take a picture). Also try the BA app, it may work when the website doesn't, or vice versa. Take a screenshot of any errors you get when trying to log in or do anything reasonable.
Try calling the call centre. You probably won't get through in a sensible length of time, but document that you made several calls. Take a screenshot and save it.

If you have made reasonable attempts to contact BA and have not been able to do so, this now puts you in a strong position to organise your own onward travel arrangements, and to claim the difference from BA later. You need to act reasonably, choose travel arrangements that are similar where possible to what you had purchased from BA, and where possible you should document with photos or screen shots that the actions you are taking are reasonable, there are no other cheaper options. Even if you don't plan to use it for booking, use a comparison service to show current market costs for the transport you are choosing. BA should respond positively to customer service claims for the cost of onward transport where you can show that BA were unable to provide you timely assistance to rebook. You should do this bearing in mind what coverage you might have from your own travel insurance policy and whether your planned trip is salvageable or if it is entirely in vain and should be abandoned. Note that BA are not responsible for consequential costs beyond their initial contract for transport, but any insurance you may have could be.
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BA have a duty of care under the UK successor to EU Regulation 261/2004 and based upon forum experience the following costs would usually be claimable from BA without any significant pushback.

£25 for reasonable meal/refreshment expenses (per adult per day)
2 reasonable phone calls per customer

If an overnight stay is required in a location a significant distance from your home address
£200 for a hotel room (for 2 people)
£50 Transport to/from the airport (round trip)

Although BA has suggested some guideline costs for duty of care as indicated above, EU Regulation 261/2004 doesn't specify any monetary cap. By documenting (take a screen capture of a hotel comparison site for example) that there were no cheaper alternatives, it is possible that claims exceeding the guideline costs suggested by BA may be met, but having additional evidence or justification for costs going over these limits would be sensible. Expenses not covered by BA may be claimable from your travel insurance subject to possible policy excesses.

Duty of care is separate from fixed sum compensation (EUR 250 to 600 depending on the flight distance) for flight delays/cancellations, which may or may not be payable under EU Regulation 261/2004 depending on whether BA can show that the flight delay was caused by 'extraordinary circumstances' and that it took 'all reasonable measures' to avoid the resulting delay. Also, airlines do not usually entertain claims for consequential losses (for example, the cost of prepaid accomodation which you can't now use), so you would need to look to your travel insurance for these costs.

EU Regulation 261/2004 does not cover delayed/damaged/lost baggage. The Montreal Convention sets an upper limit for delayed/damaged/lost baggage compensation. For more information, visit the BA.com webpage on delayed/damaged/lost baggage - see https://www.britishairways.com/en-gb...amaged-baggage

Set your expectation for the turnaround time for any refunds, claims, etc. It could be more like weeks rather than days, consider this if cashflow is concern.

You can read the forum thread for guidance on EU Regulation 261/2004 https://www.flyertalk.com/forum/brit...61-2004-a.html or check back on these forums later for more advice on claiming, but first of all look after yourself during this disruption.