The LOFAR Two-metre Sky Survey (LoTSS) is a sensitive, high-resolution 120–168 MHz survey of the Northern sky. The LoTSS First Data Release (DR1) presents 424 square degrees of radio continuum observations over the HETDEX Spring Field (10^h45^m00^s < right ascension < 15^h30^m00^s and 45°00′00″ < declination < 57°00′00″) with a median sensitivity of 71 μJy beam⁻¹ and a resolution of 6″. In this paper we present photometric redshifts (photo-z) for 94.4% of optical sources over this region that are detected in the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) 3π steradian survey. Combining the Pan-STARRS optical data with mid-infrared photometry from the Wide-field Infrared Survey Explorer, we estimate photo-zs using a novel hybrid photometric redshift methodology optimised to produce the best possible performance for the diverse sample of radio continuum selected sources. For the radio-continuum detected population, we find an overall scatter in the photo-z of 3.9% and an outlier fraction (|z_phot−z_spec|/(1 + z_spec) > 0.15) of 7.9%. We also find that, at a given redshift, there is no strong trend in photo-z quality as a function of radio luminosity. However there are strong trends as a function of redshift for a given radio luminosity, a result of selection effects in the spectroscopic sample and/or intrinsic evolution within the radio source population. Additionally, for the sample of sources in the LoTSS First Data Release with optical counterparts, we present rest-frame optical and mid-infrared magnitudes based on template fits to the consensus photometric (or spectroscopic when available) redshift.