What is the ruling regarding the Asr, Maghrib, and Isha prayers under the mentioned circumstances, and must they be made up after their time has expired?

Your prayers that you performed without Wudu are invalid because discharges nullify Wudu. It was obligatory for you to strive in searching for a place to perform Wudu, then to repeat the prayers you performed without ritual purity. Non-obligatory Ghusl does not suffice for Wudu unless you intended Wudu by it and washed the parts of Wudu in order.