From January through June 2019, Gloucester Gate will be closed to all traffic including cycles. It is possible to get through on the footway (by dismounting). We are trying to get a better solution in place but in the meantime the only other options are to use Albany Street and Chester Gate (southbound) or Prince Albert Road and Macclesfield Bridge (westbound).
Update 7/6/19: Gloucester Gate has now re-opened to two-way traffic. So cyclists can now enter and leave Regents Park here, but, unfortunately, so can drivers.