Under the lcd crack and phone rebooting after lcd replace.

Although it certainly is possible that you damaged the battery when removing it, I wouldn't necessarily assume that was what happened as the first diagnosis.

I'm not completely clear from your description exactly what's the issue with your replacement screen but it does sound like a replacement is probably going to be needed. If you can add a picture of what you're seeing on the screen, that would help.

Adding images to an existing question - iFixit Repair Guide

On many phones, yours included, there are two ways to replace the screen. The first is the one you chose where you replace the LCD by itself. You've encountered the downside to doing it that way where you risk not getting it glued on securely or correctly. The other method is generally recommended, and that is to replace the screen and frame together as a single assembly. That way the screen is already mounted to the frame so you don't have those kind of problems with cleaning the old frame of adhesive and then gluing and clamping the new screen in place.

There is, of course, a tradeoff in that you basically have to gut your phone and move all of the interior components to the new frame. The good part of that is they generally make the process fairly straight-forward and it really isn't a horrible job; certainly it's an order of magnitude easier than it would be on, say, an iPhone.

Here's an example of the screen plus frame assembly I'm talking about; this is just one of the first ones to come up on an admittedly cursory internet search; if you spend more time at it you can certainly find ones that could be more suitable to your cost/availability considerations.

Amazon.com: for Oneplus Nord N10 5G LCD Screen for Oneplus Nord N10 5G Screen Replacement for Oneplus Nord N10 5G LCD Display Touch Screen Digitizer Assembly Repair Kits with Frame Black (Negro) : Cell Phones & Accessories

I wasn't able to locate any iFixit-style written guides, but I did find what looks to be a fairly decent video showing the steps needed to replace the screen+frame assembly.

OnePlus Nord N10 5G Screen Replacement

With regard to the looping, I would suggest doing the full frame replacement first before worrying about the reboot. It's just as likely to be caused by a damaged display as it is a damaged battery, so hold off your judgement until you've replaced the screen. When you get to the part about replacing the battery, just set it in place rather than gluing it down securely just in case. Reassemble the phone far enough to be able to power it on and test it and see if it's still rebooting. If it is, you can start trying to diagnose the problem then, but there's a fair chance it won't be and if that's the case then all you have to do is glue down the battery, close up the back and call it a day.

Oh, and one final thought; in all the replacement parts I've looked at, it's rarely been more than a couple of dollars difference between the screen alone and the screen plus frame assembly, so price really shouldn't be a determining factor in my opinion.

So I'd say go ahead and do the replacement I've advised then reevaluate whether it's still rebooting. If it is, come on back and let us know what's happened and we can try and figure out your next steps. There's also the possibility that the rebooting could be caused by a connector that is damaged or not fully seated, so you could actually fix the rebooting problem in the process of swapping out the frame.

Good luck and let us know how it goes! Success stories are always good to hear, and even when things don't go right there's usually something we can learn from it. In your case you've somewhat reinforced my general advice to do the screen+frame replacements when there's a choice between that and just the screen by itself.