But Cheltenham & Gloucester-owner Lloyds did not, meaning the differential between the cost of its deals and the base rate has risen by between 0.8% and 1%.
Andy Gray, at the Woolwich, said: "We are seeing unsustainable flows of customers to the Woolwich since changes by other lenders left us with some of the only competitively-priced mortgages in the market.
"Last week we immediately passed on the full Bank of England base rate cut of 0.5%, but as a result of changes elsewhere in the market we now need to control the flow of business by making some slight increases to the rates on our tracker mortgages."
Wholesale money markets have not passed on last week's interest rate reduction, with three month Libor slightly down at 6.21%, but well above the Bank of England base rate of 4.5%.
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