The Improved Caesar Pig Latin Cipher

Ruby, 96 69 bytes

->s{/^[^aeiou]/=~(r=s.tr'zb-yadehinotu','b-zefijopuva')?$'+$&+'ay':r}

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Fun fact of the day: tr() matches strings right-to-left. I always assumed it was left-to-right.

R, 86 85 bytes

Simple way. chartr has the charming and useful property that it can specify letter ranges, which save a few bytes.

-1 bytes by stealing @GB's Ruby solution's translation strings - upvote it!

function(s)sub("^([^aeiou])(.*)","\\2\\1ay",chartr('zb-yadehinotu','b-zefijopuva',s))

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Java (JDK), 167 bytes

s->{String r="",a="aeiouabcdfghjklmnpqrstvwxyzb",c=s.split("")[0];s=a.indexOf(c)>5?s.substring(1)+c+"ux":s;for(var d:s.split(""))r+=a.charAt(a.indexOf(d)+1);return r;}

Try it online!

Credits

• -7 bytes thanks to Kevin Cruijssen, using better types.
• -1 byte thanks to ceilingcat.