Don't try too hard. This may be disconcerting, but the analogy I use is that your bag(s) are like letters in the post office sorting room. Try as hard as you might, you cannot ring up and get a postman to find your letter in the piles of mail, and put it on the next delivery round. So you best use Skype to call the baggage line between 06:00 hrs to 20:00 hrs UK time. If you can call early that may work best
0344 493 0785

You want a baggage reference number, if you haven't already, and once you have that you can change the delivery address for your onward movements. You can sometime report it all online but failures are common. It is best not be logged into your BAEC account even if given that option. It may be best to think about a time several days from now for the delivery address, rather than hope things will be moved quickly. This link for delayed baggage will take you to WorldTracer for the baggage tracking, though the interface is very poor and often unhelpful.
ba.com/baggage

You can claim essential items such as clothes and cosmetics up to £1400 per person, but bear in mind that sumi is tha absolute extent of BA's liability even if your items are never found (hightly unlikely). Note that the system is that BA will return the bags to you, you don't go to them, and so I doubt that trip to the airport will be refunded.