There are only two ways out of a kosher marriage: a kosher get or death of a spouse. (Kiddushin 2a)
It doesn't sound like any semblance of the former (a document handwritten by a Jew for the divorce of this specific couple including their names, the date, specific formulations, signed kosher witnesses, etc. presented to the wife by the husband again in front of kosher witnesses, etc.) has taken place. There's very little wiggle room with it and it is usually arranged for by trained professionals.
So they are still married. Any extramarital relations would be adultery and any progeny thereof would be mamzerim. Not a good situation.
(If it turns out that the original marriage wasn't kosher, then problem solved. But that has to be evaluated on a case-by-case basis.)