A little background on HDI.

HDI or the Human Development Index, was devised by Pakistani economist Mahbub ul Haq on behalf of the United Nations Development Programme as a means of giving countries a composite development score that took in factors beyond just national income. The first HDI report was published in 1990 (the full list of annual reports can be accessed at the UNDP site here), so it's worth noting that the USSR was only analyzed under the HDI metric for the last couple years or so of its existence.

The HDI is assembled from three other indexes: a life expectancy index (computed by life expectancy at birth), an education index (computed by average number of years of education a 25 year old adult has received), and a GNI index, based on Gross National Income per capita at Purchasing Power Parity. More information is here. Perhaps someone more versed in the metrics can speak to this, but it looks like it's measuring GNI (gross national income), not the Gini Coefficient, which is a measurement as to how inequally income is distributed in a country.

I have to say, the 1990, 1991 and 1992 reports really do not devote much space to the Soviet Union, whether existing or recently-deceased. It mostly tends to crop up in discussion of overall geopolitical trends such as the end of the Cold War.

In 1990, it gets a listing in the High Development Group with a score of .920 (there isn't a Very High Development Group in the 1990 report). This puts it ahead of much of the world, but behind Western/advanced economies, and even a few developing ones. Czechoslovakia and Chile beat the USSR out at .931 for both, and the USSR is just ahead of Bulgaria, Costa Rica, Uruguay, Hungary, Yugoslavia, and Argentina. This is probably an accurate way to position the Soviet Union in the world at the time - it was at the higher end of development, but that still put it behind more developed parts of Eastern Europe, and at about the same level as Southeastern Europe or the better-off parts of Latin America.

In 1991 it has a score of .908, and is 31st highest in the world. This put it behind Greece, Hungary and Czechoslovakia and ahead of Yugoslavia, Bulgaria, Portugal, Uruguay, and South Korea.

I guess what I would say is that the statistics for the USSR have issues, but perhaps no more issues than many countries' statistics of the time would have had. Life expectancy statistics were massaged in the Soviet period (infant mortality so many weeks after birth was classified as miscarriages, for example). Years of education probably is the most reliable statistic. GNI per capita at PPP was notoriously difficult to compute because the Soviet economy didn't function like economies with market structures, so computing a "real" score for value added was notoriously hard - Soviet economists basically used a theoretical number, since there wasn't anything like "fair market value", and non-Soviet economists could have wildly different estimates as to the size of the Soviet economy, not the least because the data wasn't transparent. You can kind of compare Soviet GNI figures in a year-on-year manner, but comparing it to other countries' economies could get weird and problematic.

With that said, the overall score, as mentioned, seems fairly reliable. Given the desperate underdevelopment that many countries and much of the world's population faced in 1990, the Soviet Union was definitely near the top - but it was not catching up with, let alone overtaking advanced Western countries, and was even falling behind other parts of Eastern Europe. It had made big strides, even since the 1970s (the 1991 report looks like it back-estimates HDI scores for every country in 1970 and 1985, and the USSR made large advances in the score over that time), but part of what that reflects is the huge windfall the USSR had from rising international oil prices after 1973. It's notable that the Soviet GDI seems to peak in 1985 and falls thereafter, which matches the fall in oil prices that occurred from 1986 on.