Lloyd's now has the right to send bailiffs round to hundreds of Names that have refused its final settlement offer.
Last November, Lloyd's won the Jaffray case, which dismissed Names' allegations that former senior members of the market had concealed escalating asbestos losses. At the time, Lloyd's said it would write off some of the £300m owed to it by more than 700 worldwide private investors and it immediately made a formal offer to its Names.
Since then, 141 individuals applied to settle with Lloyd's and 27 went through the decision panel. In total, 65 settled and an estimated £18m to £20m has been handed over.
Now the 300-year-old market has the right to pursue Names, who originally had unlimited liabilities, for money.
Lloyd's spokesman Adrian Beeby said: "We said this was the last and final offer and that is the case.
"We do not have any alternative but to move to debt collection in the future."
Beeby said Lloyd's would wait until the appeal hearing of the Jaffray trial before sending the bailiffs
to the homes of Names. The court appeal, originally scheduled for January next year, will take place on 4 March.
He said: "Theoretically, the next step would be to move to debt collection, but in the UK it is staved through the Jaffray appeal and, at the moment, we have voluntarily staved it in the US."
John Henderson, a committee member of the United Names Organisation (UNO), which has 240 members and is launching the appeal, said: "Only 16 people applied to us for permission to accept the deal. That is not frantically significant."
He said if UNO was not successful in its appeal, it would consider launching fresh legal action based on issues such as fraud and market regulation.