> In February 1943, a British bomber containing a H2S radar was shot down over the Netherlands, and the 10-cm magnetron was found intact. In short order, the secret of making successful magnetrons was discovered, and microwave radar development started.
> In June 1941, an RAF bomber, equipped with an ASV (Air-to-Surface Vessel) Mk II radar, made an emergency landing in France. Although the crew had attempted to destroy the set, the remains were sufficient for the German Laboratory for Aviation to discern the operation and its function. Tests indicated the merits of such a radar, and Wolfgang Martini also saw the value and tasked Lorenz to develop a similar system.
The secret war is a documentary about exactly that (amongst other things e.g. the magnetic mine): WW2 RADAR and radio navigation, and the intellectual battle between the Brits and Germans as they tried to locate beams and steal (genuinely!) entire German ground radar stations
It's very long, very detailed but well presented - in a style that assumes you aren't stupid.
So, you know, eat it in chunks
YouTube remembers where you left off now so it's not a huge issue in 2019
The stations of the radar network were a string of about a dozen wooden towers, each about 100 feet tall, situated along the east coast of Britain. If the German knew that they were radar stations, they probably would have destroyed them and prevented Britain from rebuilding them. Without the radar network, the Royal Air Force probably would've been destroyed in the Battle of Britain, and the Royal Air Force was the only thing preventing the Germans from invading across the English Channel.
So if it helped keep the Germans ignorant of the British radar network until the Germans gave up trying to destroy the Royal Air Force in September 1940, it was monumentally helpful regardless of what happened after September 1940.
Apart from the Royal Navy, of course. I think the general consensus now is that Germany couldn't have sucessfully invaded the UK, even if it had won the Battle of Britain?
And of course by D-Day the Luftwaffe had been mostly destroyed relative to what it was in 1940.
So, no, I tend to doubt that if the RAF had been destroyed, the Royal Navy could have prevented the German army from making a crossing of the English Channel.
Naval ships also are just a bit useful for control of the sea; indeed, historically that's been arguably the foremost purpose of the navies of island nations such as the UK and U.S.
If the Germans had tried to invade Britain, the Royal Navy would have thrown in every last ship it had to attack the German troop transports in the Channel. (I'm paraphrasing Herman Wouk here.) The RN ships would have been extremely vulnerable to U-boat and E-boat attack there, of course (just as the U.S. Navy and RN are worried about in today's Strait of Hormuz). The RN would also have been vulnerable to air attack (as the British, U.S., Dutch, and Japanese navies learned at terrible cost in the Pacific War), especially if the Luftwaffe had previously gained mastery of the skies over the Channel. But as the saying goes, desperate times call for desperate measures.
The Luftwaffe had about 2600 planes in 1940 and the ability to build as many airstrips as they pleased all along one border of the Channel. What do you think the chances were of a British ship or task force's surviving a single mission into the Channel? I agree with what you're saying that the Brits would've been willing to risk their entire Navy to stop an invasion of the homeland (which BTW was the original reason for the creation of their Navy centuries ago) but would that have been enough? (And if they lost most of their Navy, would they have been able to continue to feed their population? Remember that the Brits were dependent on food from overseas.)
If some power wants to attack a US aircraft carrier, they face the formidable task of finding it first. On the open ocean this is very hard: during the Cold War, the Soviets deployed an entire constellation of satellites for the sole purpose of locating US carriers in the North Atlantic (then transmitting the location to their submarines and long-range bombers). I mention this because the reputation for awesomeness earned by the British Navy before WW II was earned mainly in unconfined waters; against air power and subs, it is much less powerful in the confined waters of the Channel.
Also, look what happened to the British Pacific Fleet the next year when it dared to sail within range of Japanese aviation assets. And I think that happened in unconfined waters.
Honestly, I figured the GP would get a kick out of reading about a hypothetical war game of the scenario they envisioned.
Um, yes — I think I mentioned that.
From my reading about D-Day, it would have failed if the Allies had not achieved absolute air superiority. The Allied positions behind the front line were safe, and any strong points of German resistance were immediately attacked by ground attack airplanes.
The British had no reason to risk their ships, by shelling the European coastline. Little of the vital war-making industry was along the coast, and harbors and dockyards were heavily defended through geography, mines, and fixed coastal batteries.
Instead, the Royal Navy had complete sea superiority of the English channel. Trying to ship, and then support hundreds of thousands of soldiers, and thousands of tanks, on slow-moving barges, through the English channel, while being outnumbered 4:1 by the opposing Navy would have been complete suicide.
Not to mention the months of supplies, that would also have to come over the channel, that any sort of invasion would require.
If, by some miracle, Operation Sealion actually made a landing on the British Isles, the most likely outcome of any such invasion would have been another Stalingrad. 
 In one of his speeches, Hitler quite famously stated that the Sixth German Army (after its encirclement in December 1942) would never leave Stalingrad.
He was not wrong.
This is not true. The RADAR stations were well-known to the Germans, who bombed them from the very start of the BoB and devised many tricks to counter them.
By this, I mean that the Germans would've had to be very very lucky to completely stop British radar (Chain Home was pretty rubbish by the time the Germans knew about it, so it could've been rebuilt very quickly) but bombing something that small would've been very difficult given that it was in operation and well covered by fighters
The veteran British radar scientist and others interviewed on the TV show _Battle Stations_ said different.
>the infrastructure around the RADAR was more important i.e. the command centres
The fact that the stations by themselves would have been useless has no bearing on the question of how vulnerable the stations would have been to German attacks.
I didn't mention the communications lines and the command centers because they were less vulnerable to German attack.
To summarize: if they had known about it, the Germans probably could've degraded the British radar network enough to succeed in their goal of destroying the RAF. They came close to destroying the RAF as it is.
It still relied on spotters to keep track once they were over land.
Also I think a British squadron usually tried and often succeeded in meeting the German force before or shortly after it crossed the coastline, and the British squadron had radios.
Propaganda is relatively cheap and doesn't consume a lot of resources more directly useful to the war effort. You have a half dozen guys coming up with ideas and doing the graphic design, some printers, and you can either let civilians or enlisted put up the posters when they don't have anything better to do.
At that level of cost, if any piece of propaganda makes an impact, it probably pays for the entire campaign.