Nope, it's fine...as long as the new passport was issued less than 3 months before the ILR application date, it's not an issue... because essentially, you are applying to change your name on the BRP, you're just doing it at the same time as applying for a new visa.
You can just write a cover letter to explain the name change (or if you apply for ILR in person, you can explain at the appointment).
This is actually what we recommend you do if you want to change your name after getting the initial spousal visa and many people have done it this way (either before applying for FLR(M) or before ILR).
That's great; very helpful!! I wish I had known earlier.
Now another question. Can I apply for ILR on a limited-vaility US passport?
Processing time for full-vailidty at the US embassy in London is 3 weeks average, no matter if you apply by appointment or post. (But I'm going to ask again to see if there are any other options.) Only limited-validity/emergency passports are issuable right away. I arrive back in the UK on the 26th Nov and my travel plans are to leave again 27th December (though I won't go unless my ILR is settled!). It's just not enough time to wait 3 weeks for a new passport, then submit ILR application online, get a appointment (with one of the new value-added services), submit docs/enroll biometrics, and get a decision the next day (hopefully, with super priority). Even if I got a same-day decision for ILR, there's the wait for BRP to arrive.
And yet another question - Is it possible to travel on a passport recently changed to married name with BRP still in maiden (as long as it's before the 3 months are up)? If you carried old and new passports, brp, and marriage certificate or something like that. It doesn't apply to me now, but wondering if it could have worked.