Depends upon the circumstances.
If the only damage is a lost canopy and the pilot is fully in control of the airplane there really is no reason to land immediately, or even urgently. In fact, a too-rapid dash towards the illusive safety of the ground may precipitate mistakes and further problems.
I am not particularly conversant with warbirds of any era, but at the airport I’ve been frequenting this past year I do see some types flying about with open canopies on ocassion with no apparent problems, so certainly it’s not an issue for many.
You actually don’t need the leather cap and googles, although eye protection is recommended. Aside from the prop potentially kicking up debris from the ground, which will hurt like the dickens if it dinks your eyeball, the wind tends to dry out the old peepers and can make it hard to focus. This not only complicates navigation, it makes landing an adventure, too.
Lets not deal with recollections. Let’s see some evidence. Here is a picture of me prepping for an actual open cockpit flight last October. Coincidently, although that is not a P51, it is an airplane of roughly the same time period. That is all the headgear I wore, and nothing more - just a headset and some sunglasses that wrapped around somewhat. The flight lasted about an hour and a half, speeds in the 80-110 mph range (130-180 kph, roughly). I found the windshield - small as it is - did a fairly good job of shielding me from the airstream, propwash, and the odd glob of flying engine oil. I wasn’t aware of just how good until I leaned over the side while trying to get a cool in-flight picture, at which point I almost dropped the camera out of an airplane again (which is why the camera has a sturdy leash). The sunglasses - small as they are - provided adequate protection for the old eyeballs. Most of the air-pummeling I received was from air washing off the top wing, which, of course, is not an issue in a low-wing monoplane such as the P51. I could have easily tolerated higher speeds than what we actually encountered on that flight. Judging from the the picture of the P51 in question, I’d assume that the remaining windshield provides at least equivalent, if not better, protection from the airstream.
In other words, some bareheaded old fart with nothing more than sunglasses (and he can probably do just fine without those) could pilot a P51 sans canopy without significant physical discomfort. There would be more drag on the airplane, so the top cruise speed would be less than normal, it would be windy, and it would be a little chilly. That’s it. I do stuff like that for fun, and I’m a nothing-special over-40 little-girl deskjocky. A big ol’ ballsy guy pilot shouldn’t have any problem with it either.
Honestly, I felt much more beat up on a motorcycle doing 50 on the road than I felt doing twice that in an open cockpit airplane.
Well, I wouldn’t either - it’s not like ATC can do anything about debris on the ground. If it seriously affected my airspeed or an estimated time of arrival I might, because that’s relevant to flight traffic. The people I’d notify would be a mechanic (for repairs) and possibly my lawyer if the bits fell on someone who felt inclined to sue. Well, if the debris hit and then the object it impacted burst into flame I might call ATC so they could notify the local fire department, but I think in those circumstances (emergency and all) I might just fire up the cellphone and call 911 myself.
Of course, I’m talking about the US, where a lot of the time you don’t have to deal with ATC at all as a private pilot, if you choose not to. I have no idea what the regulations are in Germany, or what they might require of a pilot in a airplane shedding parts.