From what I've 'heard', there really shouldn't be much of a delay.  I've 'heard' that if two people claim the same dependent the IRS processes the tax returns and refunds as usual, but then sometime later they send notices to resolve the issue.

So *IF* that is correct, the only delay is that the second tax return needs to be filed by mail, rather than e-filing (but with the coronavirus, I suspect that processing paper returns may be slower than usual).