Context. Ellerman bombs (EBs) with significant Hα wing emissions and ultraviolet bursts (UV bursts) with strong Si IV emissions are two kinds of small transient brightening events that occur in the low solar atmosphere. The statistical observational results indicate that about 20% of the UV bursts connect with EBs. While some promising models exist for the formation mechanism of colder EBs in conjunction with UV bursts, the topic remains an area of ongoing research and investigation. Aim. We numerically investigated the magnetic reconnection process between the emerging arch magnetic field and the lower atmospheric background magnetic field. We aim to find out if the hot UV emissions and much colder Hα wing emissions can both appear in the same reconnection process and how they are located in the reconnection region. Methods. The open-source code NIRVANA was applied to perform the 2.5D magnetohydrodynamic (MHD) simulation. We developed the related sub-codes to include the more realistic radiative cooling process for the photosphere and chromosphere and the time-dependent ionization degree of hydrogen. The initial background magnetic field is 600 G, and the emerged magnetic field in the solar atmosphere is of the same magnitude, meaning that it results in a low- β magnetic reconnection environment. We also used the radiative transfer code RH1.5D to synthesize the Si IV and Hα spectral line profiles based on the MHD simulation results. Results. Magnetic reconnection between emerged and background magnetic fields creates a thin, curved current sheet, which then leads to the formation of plasmoid instability and the nonuniform density distributions. Initially, the temperature is below 8000 K. As the current sheet becomes more vertical, denser plasmas are drained by gravity, and hotter plasmas above 20 000 K appear in regions with lower plasma density. The mix of hot tenuous and much cooler dense plasmas in the turbulent reconnection region can appear at about the same height, or even in the same plasmoid. Through the reconnection region, the synthesized Si IV emission intensity can reach above 106 erg s−1 sr−1 cm−2 Å−1 and the spectral line profile can be wider than 100 km s−1, the synthesized Hα line profile also show the similar characteristics of a typical EB. The turbulent current sheet is always in a dense plasma environment with an optical depth larger than 6.5 × 10−5 due to the emerged magnetic field pushing high-density plasmas upward. Conclusions. Our simulation results indicate that the cold EB and hot UV burst can both appear in the same reconnection process in the low chromosphere, the EB can either appear several minutes earlier than the UV burst, or they can simultaneously appear at the similar altitude in a turbulent reconnection region below the middle chromosphere.