#C, 152 151 bytes
char*s="aiueokstnhmyrw";i,j,t;k(char*v){for(;*v;i?j=t:(i=t))t=strchr(s,*v++)-s;i<5?i++:j<4?j+=(i>10&i&1?i%9:1):(j=!++i);printf("%c%c",s[i],i<5?:s[j]);}
#C, 152 151 bytes
char*s="aiueokstnhmyrw";i,j,t;k(char*v){for(;*v;i?j=t:(i=t))t=strchr(s,*v++)-s;i<5?i++:j<4?j+=(i>10&i&1?i%9:1):(j=!++i);printf("%c%c",s[i],i<5?:s[j]);}