Glyphosate comes in various strengths.
I would be surpised if the usual strengths available would kill off the brambles, and suggest you dig out as many of the bramble roots with a mattock or spade before applying the weedkiller in Spring, when the weeds and brambles are really going for it.
Remember that Glyphosate is toxic and remains in the soil for quite a while and, although it is unlikely to cause you any problems, do read the instructions fully about washing hands/gloves/etc.
Good luck - we're clearing a new allotment by the way outlined above, but glyphosate @ 450 grammes per litre is going to be used after a massive attack on brambles and nettles. The Tesco glyphosate is 90 grammes per litre...