Assume you dynamically hedge the CVA from a counterparty that you have entered an interest rate swap with.
You hedge the interest rate delta with an interest rate swap.
You hedge the credit risk delta with a CDS.
What are the expected PnL from this in
1) a flat market where interest rates do not move at all.
2) a volatile interest rate market where the interest rate delta is rebalanced many times.

In the first scenario, there would be zero PnL from the interest rate swap (the hedge). CVA would be reduced to zero at maturity (assume no default). The CDS would expire with no payoff. The time decay of the CDS would be the same as the time decay of CVA, or?
In a volatile interest rate market, you would in addition buy high and sell low every time the interest rate hedge is rebalanced implying losses over the life time of the derivative.

So, how would a CVA trader try to have a positive PnL without taking directional bets? Or have I understood something? I have read a presentation saying that there is profit from the time decay of CVA (unless a default situation), bit isn’t that offset by the time decay in the CDS hedge?

CVA is strictly not a profit centre.  They add value not by increasing the numerator, but by reducing the denominator.

So your understanding and correct.

My understanding is that, the evolution of the CVA metric is non-linear and will not map on a 1 to 1 with the CDS hedge, there is a potential of the exposure to over/undershoot the hedge, the Bid-offer from the  consistent rebalancing cost will hit your pnl as it becomes too costly to price the CVA out to the client. The credit hedge will probably need a more bespoke nature to be effective. You will also have modelling sensitives in the from of Vega from implied calibration effects, as well as cross gammas also going through your pnl, these are more difficult to hedge.