The Ukraine Sponsorship Scheme allows people to volunteer to house refugees fleeing the war in Ukraine.
Home Office data from shows 32 visas have been granted to stay with hosts in the town, from 36 applications – but as of Monday, just 21 of these refugees had arrived in the UK.
Sponsors offer accommodation for at least six months, with refugees allowed to live, work and study in the UK for up to three years – but the Refugee Council said it had received reports of delays at every stage of the application process, and of documents not reaching refugees quick enough for them to travel.
Head of advocacy at the charity, Andy Hewett, called the scheme "unfit for purpose".
He said: “Responding to a serious humanitarian crisis by offering complex visa routes, putting paperwork and bureaucracy before people was always going to have tragic consequences."
Refugees have also been arriving through the Ukraine Family Scheme, which allows Ukrainian nationals to join family members in the UK.
Nationally, 107,400 visas had been issued through both schemes as of Tuesday – with 53,800 arriving in the UK by Monday.
The Home Office said some refugees are choosing to stay put or travel elsewhere.
A spokesperson called it "one of the fastest and biggest visa schemes in UK history" adding they are now “processing visas as fast as quickly as they come in”.