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A month ago I had a AA service, the bronze service. The team recommended a transmition flush as the fluid was black rather than red, so I had them do it. Now when I put my foot down harder on the gas while going slow or from stopped position (taking off from intersection etc and putting foot down to drop gears going up hill) the engine surges and only stops when I take my foot completely off the gas or put the car in neutral. In follow up with service team they recommended a new battery as it was at 60%, have new battery now but is still surging. Could it be something to do with the transmition flush? Or any other ideas what it could be?

A bronze service is a lubrication/mechanical service; it does not include a tune up. The silver service includes a tune up.
The colour of the automatic transmission fluid is really important; it is like a "health" indicator for your transmission. The brown discolouration is the residue from the friction clutches. Clean fluid is extremely important for all the friction materials like bearings, bushes, clutches, brakes and the high precisions shift valves and solenoids. It ensures long life, proper shifting of the gears and temperature control to some extent. The advice to flush the transmission was correct.

If your car is surging under acceleration, it may suffer from a misfire. Misfires are caused when the spark jumps to earth instead of through the spark plug on to earth. Some of the many causes include worn plugs, high resistance in leads or plugs, a pinhole in a plug lead, weak coil, moisture etc.
Good battery voltage is important for starting the engine but it won't be the cause of the surge.
Don't hesitate to take the car back to the garage and ask them to investigate.