First of all let's discuss why the Danish government decided to not recognise the referendum in 1946.
First it had a low turnout[1]. Next the yes to independence won by 166 votes[2], which was a plurality of the vote and not a majority. This very small plurality meant the Danish government didn't find the referendum legitimate. Iceland also had a right to dissolve the union while Faroe[3] didn't.
But let's imagine the Faroese was able to drum up greater support, so the Danish government recognise Faroese independence. If the Faroese can get a absolute majority of the electorate Denmark would likely have decided to recognise their independence. It would in general have been recognised around the world afterward.
The island have roughly 30.000 people in 1945 and the population have grown to roughly 50.000 people today. But beside that 30.000 people born on the Faroe lives in Denmark. Beside that let's be inspired by Iceland.
Iceland had Danish as first foreign language until 1999, after which it became second foreign language afterward, you can choose to learn another East Scandinavian language, but in practice people learn Danish. Because of the low Icelandic population, a lot Icelander take their education in the rest of Scandinavia, historical Denmark was the main place to study abroad and the Icelanders got special right to study there even after the union was dissolved, but today Norway have as many Icelanders as Denmark, while Sweden have a few less. The main differences between Icelanders and Faroese are that Faroe had a less developed education sector in 1944, they had less pride about their language[4], they was also far more religious and traditional. Denmark also together with USA take care of a lot of Icelandic national security. A important factor is that Reykjavik have grown since independence to have around half the population of Iceland.
So what do I think will happens? Could Faroe survive as a independent state; yes it could, but it would be far poorer.
But here we run inmto the problems Faroe need to industrialise and build up a education sector. Faroese will likely keep their favoured access to education in Denmark, so they need to send people abroad to get education. I think this will work to weaken the position of the Faroese language. In fact I think the greater (relative) poverty of the Faroese, will work to the education sector being behind OTL, and having a far greater use of Danish. We will likely see Thorshavns to large extent being Danish speaking[5].
The industrialisation will mostly build on fish, fish and fish. We may see some diversification, I could see a few shipyards being very important for the economy. We will likely also see a greater diversification of the economy to ensure a BOP surplus. But these will likely be mostly for domestic use. We will likely see forest plantage to ensure a domestic supply of timber, we will see some of the coal mines being kept open, we may see growth house growing vegetable.
Population-wise I expect the islands to have 100.000 people by modern day, most of the population increase will be in Thorshavn, which would be home to half the islands population, they will likely have a GDP per capita (nominal) around 40.000$, not bad but far lower than modern day Faroe. I could see up to a third of the population being L1 Danish speakers. It will mean that Danish are first foreign language, but we would likely see something of a Kulturkampf in modern Faroe to limit the position of Danish, with people pushing for English to replace Danish as first foreign language and pushing for greater use of Faroese in tertiary education. The Danish speakers will likely see themselves much like Swedish speakers in Finland does.
Foreign policy; Faroe won't have a army or navy leaving that to Denmark, it will be a NATO member later it will become part of EFTA, but it won't join EU out of fear of its fishing rights. It will of course be a member of the Nordic Council. The large number of Danish speakers (Thorshavn beating Flensburg as city with most Danish speakers abroad) will likely mean Denmark have a special relationship with Faroe.
[1]67,5% compared to the Icelandic 98,4%
[2]48,7% vote for independence on Faroe compared to 99,5% on Iceland.
[3]the reason I call it Faroe instead of Faroe Island are because "oe" means island(s). So I will never call them "Sheep Islands Islands".
[4]It should be said this is common, no one love their language as much as the Icelanders, not even the French. Danish was in general used as the language of public media until the 80ties.
[5]Likely using Gøtadanskt/Gøta ("Street Danish" in Faroese), a local variant of Danish used by Faroese speakers, Danish are also today spoken as first language by 5% of all school children and a unknown number of adults. Example