Thank you all for your answers. I let curiosity get the better of me, and actually tested it for myself (which I was trying to avoid in the first place). I hope others find these results useful.
As you can see, with the exception of IE, Firefox and Chrome will warn about weak hashing algorithm usage, but completely ignore the issue of 1024bit keys.
Tested July 3, 2017
Case 1: 1024bit key, MD5, no subjectAltName
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Firefox = Warning that cert is hashed with disabled algorithm (MD5)
Chrome = Warning that cert is missing subjectAltName
IE = no warning
Case 2: 1024bit key, MD5, subjectAltName same as subject CN
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Firefox = Warning that cert is hashed with disabled algorithm (MD5)
Chrome = Warning that cert is hashed with weak algorithm (MD5)
IE = no warning
Case 3: 1024bit key, SHA256, no subjectAltName
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Firefox = no warning
Chrome = Warning about certificate missing subjectAltName
IE = no warning
Case 4: 1024bit key, SHA256, subjectAltName same as subject CN
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Firefox = no warning
Chrome = no warning
IE = no warning
[1] Browser versions tested:
* Firefox 54.0.1
* Chrome 59.0.3071.115
* IE 11.0.9600.18698
[2] All tests performed on Windows 8.1 32-bit OS
[3] All server certificates signed by a self-signed Root CA,
with no Intermediate CA. Root CA certificate was installed
into the Windows certificate store using mmc.exe, and
Firefox using the Options GUI.