Indeed, you can have some nice diving in Romania, You are British, so you aren't afraid of cold water or poor visibility. If you are into wreck diving, you'll find some interesting sites: two Russian submarines (ShCh-213, infamous for sinking the Struma and a yet to be identified Malyutka class) and a Russian destroyer (Moskva). There is also a cargo ship from WWII - Arkadia. We also have some modern cargo ships, out of which 3 are in the close vicinity of Constanta harbour - they all sank by hitting the huge harbour breakwater in storms. And there are some more, but those are the most well known.
Most dives are in the range of 20-26 meters. ShCh-213 is up to 32 meters, Moskva is at about 43-45, and the Malyutka one at ~37 meters. Except for the submarines (which have the hull completely closed), there are some nice penetration oportunities if you are into that. All year round, the water at those depths will be between 7 and 12 degrees. Surface water can reach up to 25 degrees in summer, but there will be a thermocline at 7-10 meters normally, and then you'll drop into cold waters. Most of the time the visibility under the thermocline is very good on the wrecks out at sea (shch2-13, Moskva, Malyutka) - usually more than 10 meters, a little less better at Arkadia. The worst is at the breakwater wrecks (Paris, You Xiu and Sadu). In the worst case you can have a 1-2 meters visibility, but statistically over 8 years of diving there, about 60% are good visibility dives according to my logbook.
I personally prefer the Aquarius dive center from the links above. With them I have dived to all those sites. They have a fast (200HP) RIB, the travel to all the dive sites is between 20 minutes and 1 hour to the most far away. The owner is a Dutch guy who moved to Romania a long time ago. They are also great into servicing equipment, and they can provide PADI and IANTD training. The drive from Bucharest is 2 hours to Constanta, so most of the times I leave at 6AM from home, do the dives and return in the evening. Their boat will be in the water this year in the second half of May.
As a final tip, follow windguru.cz for marine forecasts. When you see that the South wind has blown in the last 5-6 days be very happy, it brings crystal clear waters. When you see the North blowing, chances are the water will be murky from the Danube.