The water sitting at idle in the coil is at whatever temperature the boiler happens to be, which is "hot enough". When the flow starts that first slug of water is hotter than what immediately follows, since the slug of cooler water doesn't heat up to the full boiler temp on it's way through. As the burner kicks on the boiler temp rises enough so that the output of the coil is at a decently hot domestic hot water temperature. If the low-limit on the boiler isn't high enough or the coil is liming up a bit that delay will be worse.
Most tankless coils can deliver at least OK hot water performance at 150Flow-limit when new, but may have to be bumped up to 160F or higher over time (and you say this one is "old".) What are your aquastat settings?
Bumping up the temperature may be a reasonable short-term fix, but it also increases the standby losses of the boiler, for lower net efficiency. It may or may not be worth de-liming the coils. Sometimes de-liming uncovers pin-hole leaks that had been scabbed over and sealed by lime deposits, causing the system to slowly over-fill until it's in an over-pressure condition, but if you want to give it a shot, be prepared with a "plan -B" if that happens.
Plan-B would be to replace the coil, but that's not necessarily the best solution if you'll be living there for years/decades. An indirect hot water tank operated as the priority zone off the boiler allows you to lower the idling and average temperature of the boiler, lowering the standby losses, and if coupled with a heat purging economizer control can lower the number of ignition cycles and overall burner-on time. Those controls still have some benefit even when using a tankless coil for the hot water, but the idle temperature would still have to be high enough to deliver reasonable hot water performance with the coil, rather than heat purging the boiler down to 140F on heating calls before engaging the burner.
