Definitely first port of call is your network, ask for a signal booster device that you can plug into your internet router at home. Vodafone's signal booster product is called SureSignal I think. If they try to charge you, make a bit of fuss on the line (threaten to leave, say its unacceptable there is no coverage in "London") and they will likely give it to you for free.
Otherwise the only way really is to test out all the networks. Go on eBay/Amazon and purchase PAYG sim cards for each network - should be 99p or less delivered each. And then test them one at a time in your phone.
Note that it could be the phone. Especially with 4G and especially if you bought your phone overseas. Not all phones have the full spectrum of 4G bands. It is a complicated subject but Google can provide more info