The effect of Brexit on the construction sector in the UK could be enormous in some respects and negligible in others.
International labour force
The construction sector is hugely reliant on skilled immigrant labour from the EU, perhaps more than any other sector except agriculture. In the absence of free movement of workers from the EU, the cost of construction in the UK will rise. In addition and perhaps even more significantly, the UK's infrastructure pipeline will become impossible to deliver because demographic changes in the UK mean that there simply are not enough skilled workers entering the construction workforce. Could the sector have delivered the 2012 London Olympics without this labour? Will the sector be able to deliver future mega-projects, like EDF's new Hinckley Point nuclear plant?
Design and manufacturing standards are international. Construction materials imported into the UK will almost certainly comply with EU standards and construction materials manufactured in the UK will need to comply with EU standards if they are to be exportable. Similarly, design standards are international. The UK has been working through a process of conforming British standards to the EU's Eurocodes, eg British Standards for the design of structures implemented the Eurocodes in 2010. While there may be no legal reason for the UK to continue to implement these standards, it is difficult to see why the British Standards Institution would seek to turn the clock back or allow the UK to fall out of step with the rest of Europe.
From a purely legal perspective, the impact of Brexit will be less apparent and certainly less immediate. EU legislation can be directly effective but in the UK it is almost always implemented via UK legislation. The UK is one of the most compliant states in the EU so the UK will need to actively repeal or amend UK legislation implementing EU legislation before the law in the UK changes. To take a specific example, the Work at Height Regulations 2005 implement European Council Directive 2001/45/EC. It is unlikely that the UK will devote resources to amending this.
Legislation yet to come
Following Brexit, the UK will need to negotiate a series of bilateral trade agreements with the EU in order to access the Common Market. The experiences of Norway and Switzerland suggest that these agreements will require the UK to implement a great deal of EU legislation in any case (without the benefit of having had a hand in writing it). It is therefore likely that not only will the UK retain existing EU legislation, it will be obliged to continue enacting it.