$ cat upd.awk

BEGIN { ORS="" }
TAG=="NAME" {
        name=$2;
        gsub(/(^[ \r\n\t]+)|([\r\n\t ]+$)/, "", name);
}

(name=="XXX") && (TAG=="EVENTTYPE") && /Delay/ { $2="Down"; }

{        printf("<%s",$0);}

$ awk -f yanx.awk -f upd.awk input.xml

<name>XXX</name>
         <eventType>Uptime</eventType>
         <eventType>Down</eventType>
         <eventType>Down</eventType>
<name>YYY</name>
         <eventType>Uptime</eventType>
         <eventType>Delay</eventType>
         <eventType>Delay</eventType>
<name>ZZZ</name>
         <eventType>Uptime</eventType>
         <eventType>Delay</eventType>
         <eventType>Delay</eventType>

$