I wouldn't expect issues in mainstream browsers. They tend to get intermediate updates in a separate way from the browser vendors. (See Firefox's blog post on how they preload all the intermediates by pulling from CCADB.)
Other uses (API calls, Java services, etc.) might be a little more complicated to predict. Way back when DST Root CA X3 cross-signed Let's Encrypt's intermediates, that root had an issue with not updating its published CRL and some systems returned very cryptic errors that took a while to decipher. The thread is actually really confusing since people were conflating issues that SSL Labs reported that were actually on their end, but there were actual error reports sprinkled in there as well.