I think it is unlikely for the reasons given below. Despite already-existing automation of legal tasks, the demand for lawyers, the work available for lawyers, and legal fees are all on the rise.

It's interesting to look back to the legal market of the early- to mid-1990s, when the market for new lawyers and lawyers in general was soft, and then the USA experienced the tobacco litigation and Microsoft antitrust litigation. This soaked up thousands of lawyers working "overtime" and saved the doomed careers of so many junior to senior lawyers who were un- or underemployed. Nowadays, with ML-based data processing and database/search tools, it's not inconceivable for a handful of lawyers to handle this work.

A smart person, whose name I don't recall, analyzed whether lawyers will be replaced by technology, and broke lawyers' roles into four categories:

1. PROCESS - Drafting contracts, preparing other paperwork, filing things. It is conceivable that a lot of this can be replaced, but (a) clients have a knack for doing things that don't fit into a mold, (b) contracts need to get revised, which is usually too messy for AI/ML, and (c) the lowest level and lowest paid attorneys are still miles ahead of AI/ML on this front, so there is not a big cost savings. This is viewed as a necessary training ground for irreplaceable (see below) senior lawyers. Robots can play soccer just as poorly as children, but if you replaced children with robots, then you would never get your next Messi.

2. ANALYSIS - Reviewing and understanding changes in codes, statutes, regulations, and case law, plus their implications for the future. You could also include writing those codes, statutes, and regulations. This function will not be replaced by AI/ML in our lifetimes.

3. ADVOCACY - Appearing before courts, tribunals, regulators and administrators, arbitrators, and other forums, and even government clerical workers to advocate for clients. This includes the judges themselves and their staff. Obviously this human element will not be replaced.

4. COUNSELING - Listening to and understanding the client’s needs, knowing the pathways, strategies, and tactics to get your client to achieving their goals, and advising them. This ranges from merely being a person to whom the client can vent all the way to the thousands of dollars per hour lawyers who devise strategies to eliminate tax and liability for mega-corporations and billionaires. Almost no human wants to tell their story to a robot. No AI/ML in our lifetime is going to be able to understand what another human’s needs are, ask the right questions and elicit the needed responses, while showing empathy. No AI/ML in our lifetime will have the trust of a CEO or senior manager.
There is a fifth function that smart person didn’t consider:

5. “INSURANCE POLICY” - Hiring a lawyer wins you the right to sue them for malpractice if they screw up. Because of this, lawyers are very thorough, cautious, and conservative, often vetting untested ideas and risk-carrying work through many layers of their law firms. Successful legal malpractice suits are rare, but the threat of them and the reputational damage makes legal malpractice itself very rare.

Could a robot replace immigration lawyers? Maybe in the PROCESS aspect of things (filling out the forms). After all, TurboTax is perhaps already as good as an accounting firm for the vast majority of Americans’ taxes, and you could perhaps say the same of Willmaker Pro compared to an estate planning law firm for most Americans. But that seems unlikely, because, unless the mechanical process of filling out a tax form, you have to interview the people, who want a human to listen to them, understand the employer’s needs, and come up with a strategy. The client wants a human to explain it to them and comfort them as to their odds. If the AI/ML screws up, only a human can fix it (low probability) at multiples of the cost of hiring them in the first place. Remember, immigration law is literally a “bet my life” proposition. Do you want to save a few $$ hiring a robot?

So it seems unlikely to me to see, in our lifetime, robots replace lawyers on any meaningful scale, whether in immigration or any other field of law.