The company who put the 'black mark' on your credit file is the only one who can remove it.
If this is all that's stopping you from your dream, swallow your pride, contact the mobile phone company and ask them
very nicely to remove the details from your credit history.
They may ask you to pay what they think you owe them. Try to negotiate - appear willing - some money for them is better than no money. Remember, they are holding all the Aces. You need to give them what they want, not the other way around.
I have a niggling feeling that the CCJ info comes directly from the Court Service, in which case you need to get that shown as 'settled' rather than an outstanding CCJ.
Go to
moneysavingexpert.com and follow the advice for repairing and improving your credit history.
Hope this is useful.