I believe Google has just recently discontinued support for the older YouTube app on the 2010 iPad.
You CAN still access YouTube website from a web browser.
Use the mobile version of the website, though. The mobile website of YouTube works better on the 2010 iPad.
Also, if you can, DO NOT use the iOS 5.1.1 version of the Safari Web browser, if at all possible.
It’s 7 years old and, probably, not safe or secure enough to use.
There are a few third party web browsers that still supported iOS 5.1.1 for quite a few years after iOS support ended for the 2010 iPad.
To get newer apps for that iPad, if you have a computer with an older version of iTunes (version 12.6.3 or earlier ), you can try downloading the recent app/s, by accessing the iOS App Store, through iTunes, and then, download the recent version of the app/s you like to use on your iPad 1, to iTunes on your computer first.
IF you own a computer with an older version of iTunes already on it, try using that version as an old 2010, iPad 1 does not need the latest iTunes version.
If you have a computer, but NO iTunes on it, here is a link to the older 12.6.3 version for download found in this Apple weblink.
Older versions of iTunes found here.
https://support.apple.com/downloads/itunes
iTunes 12.6.3 here.
https://support.apple.com/en-gb/HT208079
DO NOT download the latest version of iTunes (version 12.7.x) as there is no more support for saving apps in iTunes under the most recent versions.
If you have already downloaded the recent 12.7.x version of iTunes, NONE of what I have wriiten here in this post reply may work!
Then, back on your iPad 1, you can try to download the latest app from the iOS App Store and if the developer/s has/have left an older, compatible version of their app/s on Apple's app servers, then you will receive a small window pop-up prompt that states that there is an older version of the app that is compatible with your device.. Would you like to download this version now.
Tap the OK button just below the prompt.
Third party web browsers that should still have an iOS 5.1.1 compatible version available.
Most all of these are on my 2010 iPad,currently.
Perfect Web browser
Dolphin Web browser
iCab Mobile Web browser
Maven Web Browser
Opera Web browser
Maxthon Web browser
FYI,
Not every app you attempt to download will have an iOS 5.1.1 compatible version. For older apps that may still have an older iOS version available, the lowest compatible iOS version of an app is iOS 4.2.1.
The search for iOS 5.1.1 compatible apps will be trial and error, but they are out there.
This website may help you locate the apps you can use with your original iPad 1.
http://findoldapps.com/us/any_category/5.1.1/ipad/updated_date/desc/30/1
My 2010, iPad 1 has over 100 apps on it that are still functioning, but older versions of apps that I currently run on my newer iPad models.
Good Luck!