As many Germans fled Nazism, did some spies mix in?

I think you are right that a closer look at the makeup of the 33-member Duquesne Nazi Spy Ring might be instructive here.

While every member of the ring had pre-existing ties to foreign countries^*, not one of them came to the USA as a refugee from Germany. Several in fact didn't come from Germany at all. The ringleader was South African.

There were in fact several people of Jewish extraction who were convicted of spying on the USA during this period. However, what they tended to have in common was that they were not themselves refugees but rather descendants of refugees, and they spied for the Soviets (their enemies), not for the Nazis.

There was at least one Nazi spy caught in England posing as a refugee from an occupied country, Willem Ter Braak. Presumably several more spies caught early on in their (often comically bad) infiltration attempts were planning on doing the same thing. Of particular interest would be Jose Walberg, Karl Meier, and two other Dutchmen (whose names I couldn't dig up), who again were comically incompetent, but had forged Dutch refugee papers on them. This spooked the British enough that they detained and started screening all Dutch refugees.

There was another set of refugee Nazi spies in England who, as per the pattern in the USA, were not refugees from Germany, but rather from the USSR (their enemies). Presumably they acquired their love of Nazism from their hatred of the Bolsheviks. Examples of this class are Marina Lee and Vera Von Schalburg

^{* - The exception here is one Kansas native who was the ringleaders girlfriend, and was convicted basically of not reporting on him.}

I think if one is to take a lesson from all this, its that refugees themselves don't tend to be a danger, outside of spying for the enemies of the regime they fled. However, the existence of a refugee community does provide a tempting bit of cover for those who might try to sneak in through other means. The really big danger (such as it is) seems to be in your second-generation citizens.

There were many people fleeing the Nazis in the 1930s, a lot of them Jewish. Mostly these people were welcomed by Britain. However what greatly complicated their assimilation into British society was that a lot of them were German speakers. Passports did not record people's religion, merely their nationality. So for official purposes, fleeing Jews from Vienna, were recorded as Austrians, and considered German. Often they had to explain to neighbours and others that they were Jewish and hence refugees. Equally among Czech or Polish refugees there may well have been people who were sympathetic to Hitler. Generally so far as the public were concerned anyone speaking German was suspect.

After war broke out parliament gave the government powers under the Defence of the Realm Act, to intern aliens. Anyone who was not British, (in practice mostly Germans, Austrians and Italians) was forced to register with the police. Their cases were considered and depending on what sort of threat each appeared to pose, they were categorised. The lowest category were free to live among the population and merely report at intervals to a police station. They were generally not allowed to live near the coast. However the highest category were put in internment camps for the duration of the war. To begin with, due to the pressures of events, all this was quite crudely done. Some Jews found themselves in internment camps where they were living cheek-by-jowl with Nazis. People who were seen as a risk were sent to such camps as the one on the Isle of Man, out of harm's way.

There is a considerable literature on this subject, but I am not really expert enough to point you to it.