Firstly is this a start up after a long period of non-use? That allows the oil to drain down from the top of the cylinders into the sump, and hence it does take some time to build up pressure and fill the hydraulic tappets. Some of the early Rover V8s required priming the oil pump after a rebuild, but I think your car would be after these (assuming the engine is original)
Also the grade of oil can affect how quickly the tappets fill up.
If I am not mistaken you engine will be one of the low pressure/high volume Rover V8s which require lots of flow but relatively little pressure, so again it may take a little longer for oil to reach the tappets.
However none of the above explains why it has just started happening. It is possible that a tappet (or a cam lobe) has worn badly and this is now starting show. Do you know the history of the car , has it been regularly serviced , is the oil clean and does the filter look new?
The noise does sound very loud for a tappet, so I would check that there is nothing else wrong.
