Hi Lynda2x! Don't panic, worry, you'll be just fine and everything will workout!
Sorry for the change of plans with being able to stay w/ the family and all, but I'm a firm believer that apparent bad situations can lead to great/good changes! So be hopeful!
So, go, you'll get there, and be w/ the family for 1 week, it's better than nothing. Start looking for work right away. Do you 'have' to work? For how long were you planning to travel? What's your budget now? How long will it allow you to travel if you don't find work?
After the 1 week, unless things change, you'll have to stay at a hostel most likely. They range from 13-25 euros. Are you going anywhere else besides London? Be sure to research some cheap hostels in London before you arrive there so you have some options in the beginning.
If you travel to other cities/countries, besides the TP and HI online hostel search engines, tourism offices at major airports and train stations are great resources as well. As long as you go during business hours, they can help you find hostels as well.
What's the worst case scenario? You run out of money and come back earlier than you anticipated - that's no sooo bad right? You'll still have traveled, so don't get so stuck on the fact that things took an unexpected turn, just try to make the best of the current situation.
And remember, we're here for you.