Hi Ed!
5 suitcases plus a telly would probably fit into my estate car. So the cheapest way of shifting your gear might be to find a mate who's got an estate car (or, of course, a van). It's likely that you, and your girlfriend, would have to travel separately (either by public transport or with your mate making two trips).
Next cheapest (assuming that you don't drive yourself, which I seem to recall is correct) would be to get a mate to hire a transit van. (A car-based van might do the job, and be slightly cheaper, but transits usually have 3 seats up front, which makes it easier for you to accompany your gear).
Next after that, would probably be to look for a 'man and a van' service advertised in your area. A bit of googling suggests that T&S Transport might offer the type of service you require. They're just down the road from you. (Caxton House, Old Station Rd, Loughton - 020 85025602).
A black cab would be very expensive. However, you might find a local private hire (minicab) company who'll give you a quote for a driver with an estate car. (You'd probably need to ensure that, say, your partner was already at the new address, waiting for the car, while you loaded everything at the old address. Then, if you didn't fit in the car yourself, and had to use public transport, the driver wouldn't be kept waiting upon his arrival).
Chris