The capillary filling speed of wetting liquids of varying viscosity and surface tension in hydrophilic nanochannels with an elastic capping layer has been analyzed. The channels, with a height just below 80 nm, are suspended by a thin flexible membrane that easily deforms due to the negative pressure which develops behind the moving meniscus. In the elastocapillary filling of the channels, two opposite effects compete: the decreased cross channel sections increase the flow resistance, while the Laplace pressure that acts as the driving force becomes more negative due to the increased meniscus curvature. Although the meniscus position shows a square root of time behavior as described by the Washburn relation, the net result of the induced bending of the membranes is a definite increase of the filling speed. We propose a relatively straightforward model for this elastocapillary process and present experimental results of the filling speed of ethanol, water, cyclohexane and acetone that are found to be in good agreement with the presented model, for membrane deflections of up to 80 percent of the original channel height.