The URL to tap on (or type in) in Edge is web.telegram.org. Of course, you can turn this into an 'app' yourself with PAWA or just pin it 'as is' on your Start screen. It's up to you.
Here's the Telegram PWA in action:
Signing in is by phone number in traditional fashion and is seamless, with a few seconds taken to 'generate keys', the most time consuming part of the encryption process, so it's handled up front; (right) signed in and chatting away. Here Steve Heinrich is typing and you can see that in real time - all very impressive for a PWA.
The hamburger menu is top right and lets you find specific people, etc. (right) in Settings, you can ignore the 'notification alerts', since these won't work under Edge in Windows 10 Mobile - sadly. So the PWA is not ideal for background operation, its biggest limitation.
Still, when in the PWA, you can chat away as if in a fully native client application - it's slick and works well.
Comments welcome - I'm sure we have some Telegram users in the AAWP community - how does this stack up for you under Windows 10 Mobile and Edge? There's also the third party scene, of course, with Unigram X UWP being the client to go for. Begging a comparison from Telegrammers! (is that a word?!)