Nothing herein contained shall require any change in the plans, construction or designated use of a building under construction, provided that at the effective date of this chapter the construction shall have proceeded to the completion of the foundations and the setting of the first-floor beams, and provided, further, that the work thereon shall be diligently prosecuted to completion within one year from the effective date of this chapter. The Board of Appeals may, in its discretion, due to the size or nature of the building or any special circumstances which, in its judgment, are sufficient, extend the date of completion beyond the one year above fixed.