I flashed 0.14.xx and 0.15.xx to my ESP8266s,
I tried it with D1 Mini, NodeMCU and Amica, all from AZ Delivery.
All of them are working in AP mode and also connecting to
the router after set Wifi settings , but no WebUI available,
neither in the browser, nor in the app. I used ESPTool as
well as the web flasher for upload the image. If I downgrade
to 0.14.0 everything is working fine again.
What am I doing wrong?
Some poor quality ESP devices are incapable of handling increased power draw from CPU.
You can try adding a capacitor to 3V3 regulator to keep power supply a bit more steady during boot time.
Or just replace those ESPs with newer ones (ESP32) if you need newer version of WLED.
I have many of the AZ Delivery ESP8266 running different tasks in my home, never got a problem with it. I got some ESP32 as well, but a 8266 should be capable and enough for WLED, or not? What changed in WLED from 0.14.0 to next version? Is there anybody else having these problems with 8266s?
Yes someone reported the same issues few days ago , i no longer use esp8266 in production but to test I used D1 mini . Flashing with OTA the github CI bins located here Merge pull request #4060 from DedeHai/0_15_bootdelay · Aircoookie/WLED@42decbd · GitHub both firmware-nodemcuv2 and firmware-nodemcuv2_160 worked without issues .
Next I tried WLED_0.15.0-b4_ESP8266.bin from the release page
Releases · Aircoookie/WLED · GitHub and also did not face any connectivity issues .I should note that I reset the units after each OTA ( from setting /security page )
The one thing I see different is that the bin from Releases · Aircoookie/WLED · GitHub after I reset the unit had the Disable WiFi sleep unchecked so might be try that ( it did not make a difference in my tests but might be that could help with your connectivity issues ) .
@blazoncek Is there a reason that Disable WiFi sleep is unchecked in the release page if that was intentional
I already tried this, but still no WebUI.
None of my ESP8266 have issues whatsoever with WLED (and I have in excess of 50). Any version of WLED.
Though, there were some people complaining that (mostly old or cheap Chinese clones) have issues with recent WLED.
What changed? If you did not notice (or read release notes) quite a lot. 
WiFi sleep has an unfortunate effect to postpone delivery of UDP packets which is very important for audioreactive sync.
Which chips are on yours, to get next time better ones. 
Mine are all AZ Delivery labels.
AZ Delivery is just a brand that repackages cheap clones.
I have a few ESP32 from them and use them for development purposes only as they have slow flash and mediocre performance.
Out of the myriad of dev boards I would not look any further than those made by Lolin or LilyGo. Some say that Xiao is also good, and M5stack as well. Testing TenstarRobot ATM but undecided yet. There are also several independent DIY projects and you can look them up on official KB.
The two esp8266 ( D1 mini ) that i was testing with did not also face any issue on latest , the wifi sleep is just noted as it might help . I had to dig one more and flash it and also did not face any connectivity issues . All my esps are from aliexpress and I only select the cheapest
I solved it! 
As with many other IoT things, it was the Wifi channel!
Explaination:
My router is running on channel 13 in 2,4GHz, because of to much other Wifi on the
channels below. As I tested everything else and cant belive that the cheap chip is the fault,
I changed my channel to 11 and there it is, now everything is working fine again, WebUI
is working very fast. So there must be a change of some Wifi settings from 0.14.0 → 0.14.1.
Maybe someone else can prove this behavoir with channel 13.
I got it so often with sonoff and all the IoT things, that I am ashamed to get there only now. 
Edit: I tested again, if I switch back to Ch. 13 → no WebUI from all versions above 0.14.0.
Switching back to Ch. 11 (or below) everything works fine.
Edit II: In former times, it was always some sort of country settings in the Wifi of ESP FW,
settings set to EU or germany solved the Ch.13 problem.
Interesting find , I would have to keep a record of that to test if someone else would be having the same issues as that could explain why these issues were reported at random and only with few users . Not sure if those issues were limited to AZ Delivery ESPs and/or only on EU .
I checked my router setting and the ch was set to auto ( Ch 4 is auto selected ) and I am not in EU
Please add this information to Github issues opened.
Is there already a issue opened with this topic?
I think there many but found one here 14.1 can't connect to router on 14.1 on ESP8266 Wemos D1 mini · Issue #3690 · Aircoookie/WLED · GitHub
This even included a suggestion from blaz to swap channels .
Might be opening a new case marked as solved could be best so others could see easily that but its blaz call . In all cases if you could include as much info/logs /screen shots then might help for future cases as unfortunately i could not replicate that but we will be setting up a separate router so we could easily test with different channels
But why this isnt an issue pre 0.14.1, i would look for that cause rather. 0 issue with 0.14.0, full time issue with 0.14.1 onwards.(i have not tested the latest)
If staying on older builds works for you then why not ( in some cases there is really no need to update ) .On the other hand esp8266 has been totally dropped from other projects so in wled I think it takes a lot of effort by mainly one person to keep it alive .
All of that with the note that not everyone is having issues with esp8266 connectivity ( i personalty no longer use esp8266 in production as it is not ideal even with other projects if you have a demanding requirement but connectivity wise we seen no issue on latest wled builds)
If you have free time you could look at code and compare to see what exactly changed and good luck with that .
Please check release notes. You may find something.
Yes, i would simply drop support for esp8266 and avoid anger and frustration from both the user and people trying to fit the esp8266 going forward, wled is not getting any lighter in the future i assume?
Alternatively make an official light version that any esp8266 can use without any “hacks” made to the firmware. Getting enough power and fitting the firmware onto the thing is escalating only further. Finding these hacks must be an effort to itself.
I have 2 chinese D1 Mini copies(assuming theyre copies although they said lolin) that worked with 0.14.1, theyre still the same devices i have running wled in the house, all but those 2 went into this weird behavior that seemed like boot looping or crashing in few minutes when powered. Ordered from aliexpress one time purchase, should be the same batch too, ams1117 regulators.
Im too lazy to solder better caps onto the boards, or to do other modifications to them. Or to find out how to compile a lighter version of WLED myself. Although i could be interested in this compiling stuff if it doesnt take too much of knowledge. So far ive only compiled ESPhome firmwares as thats how it works and a linux with instructions once.
EDIT: Im just slightly mad because my gear worked fine as it was for years and out of the blue comes an update where it no more was. Never even thought of using these for anything other than basic indoors and balcony colored lighting together with automations and HA scenes. I’d just like to get a clear answer if i must get some esp32 to continue. Its just drilled back to my head to update devices to the most current firmware because of possible security risks, wled is using my internal wlan so i assume it also poses a risk. Its a big job to undo all the custom locations ive installed the old ones to.
I am fine now, everything works like it should.
Many thanks to @blazoncek & @ALDIY! 