How long will it take a baby bearded dragon to outgrow a 34 gallon uk tank?

It depends what the dimensions are - not the capacity ! Bearded dragons can grow to 18 inches as a fully-formed adult. Their vivarium at that size should measure 36 x 18 x 18 inches as an absolute minimum - to give the lizard enough space to move around and regulate its own temperature.

Your measurement of a '34 gallon tank' - implies you're using a fish-tank. Fish tanks are completely unsuitable for reptiles - as the majority need extra heat sources to raise the temperature of the enclosure to the correct level. An all-glass enclosure will simply allow about 70% of the heat generated to escape - making it more expensive to heat, and (more importantly) putting unnecessary strain on the heat-source and thermostat as they try to maintain the set temperature !