The value of properties in southern parts of Scotland, which include the two areas, plummeted by 8% in 2012 and there is little scope for a recovery this year, said the Nationwide Building Society.
The drop was only beaten by a fall of 9% in Bradford.
Northern Ireland had the biggest falls, with figures of 10% recorded in County Armagh and County Down.
Overall, prices in Scotland fell by 3%, meaning the average price of a home is £131,795.
However, Aberdeen, which has been boosted by the strength of its oil industry, was named as the best-performing area north of the Border in 2012, with prices rising by 1%.
David Strang Steel, partner in Strutt & Parker's Banchory office, said the Nationwide's estimate for the area was lower than he would have expected.
He said: "Certainly, rural prices have risen considerably more than 1% and the property market in Aberdeenshire – particularly for those houses within 20 miles of Aberdeen – remained extremely strong in 2012, underpinned by the buoyancy of the oil industry, high employment and the romantic, historical and royal associations of Royal Deeside.
"Even in November, at what would normally have been a fairly quiet time of year in the property market, we had five or six successful closing dates where the properties all sold for above their asking prices."
He added: "The average sale at a closing date in Aberdeenshire achieved upwards of 10% above the asking price last year. Correspondingly, rental prices have also gone up – often by more than 10%.
"The 3.3% fall in average prices across Scotland is skewed by the west and south, whereas Edinburgh and the central belt remained fairly static last year."
House prices in England outperformed the rest of the UK in 2012, while Northern Ireland's falls have been put down to a readjustment of the market there following a boom before the financial crisis.
Nationwide's study found evidence the north-south divide in England is widening, with the price of a typical home in the south at a new high of around £95,000 more than in the north, representing a 2% increase compared with the end of 2011.
Across the UK, house prices fell by 0.1% month-on-month in December and they are likely to remain flat or edge lower during 2013, Nationwide predicts.
The monthly decline meant that at £162,262 on average, prices dropped by 1% over 2012, reversing a 1% increase in 2011.
Sharp contrasts in the performance of the housing market were highlighted across the UK, with prices rising by 6% annually in Cambridge, compared to the 10% falls across County Armagh and County Down and 9% fall in Bradford.
Prices in England fell by 0.4% over 2012 to reach £186,390, making England the most resilient of the home nations.
Wales saw a 2.7% annual fall in prices to reach £131,630 on average, and prices also dipped by 3.3% in Scotland to £131,795.
Robert Gardner, Nationwide's chief economist, said the market is likely to remain sluggish amid the difficult economy.
He said: "With the economic recovery expected to remain fairly weak, the housing market is likely to be characterised by low levels of activity in 2013, with prices remaining flat or modestly lower over the course of the year."
Eleven out of 13 UK regions saw annual prices fall during 2012. London recorded the strongest increase, with prices up by 0.7% at the end of 2012 compared with a year earlier.
Prices plunged 8.2% in Northern Ireland year-on-year to £104,282, leaving them more than 50% below their 2007 highs.
Recent lending figures have shown a pick-up in mortgage approvals to home buyers and the Council of Mortgage Lenders has said it expects the housing market to "feel more stable and positive" in 2013, amid Government schemes to boost lending.
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