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Right now I’m applying for a firm that I’m currently on a deal accross from. Will that create any issues with me clearing a conflicts check? And will that create any issues with me getting an offer?

Sounds like a conflict to me, definitely can have the appearance of one. Although I am sure you would be fighting for your client, it could look like you give the other form a bit of an advantage or a better deal for then to land yourself a job.

So what would be the result here? Can it be resolved with just putting up an ethics wall? Or would they have to delay (or forgo) hiring me?

I would think they'll just wall you off. The only bigger conflicts issues I've seen are when you've done repeated work opposite from one of the new firm's main clients, and folks are unable to get a waiver from the client. For example, at your current firm, JPM is a huge client of yours and you've done a ton of work for them. But at your new firm, you have a new client that works across from JPM on almost all of their deals. You can get walled off, but doing so may limit your productivity at the new firm. I think some big institutional clients just don't do waivers or have weird, strict guidelines (like they require a 4 year gap between when you represented them previously and when you can be adverse to them).

Anyway, don't let it stop your application. Just disclose to your new conflicts department if/when you get an offer and they can sort through it.