When a person relieves himself, he must cleanse himself of impurities either with water – which is best and most perfect – or with something other than water which will remove the impurities, such as toilet paper, fabric, stones or something else.
Shaykh Ibn ‘Uthaymeen said:
When a person relieves himself, he must do one of the following three things.
1 – He may purify himself with water, which is permissible. The evidence for that is the hadeeth of Anas (may Allaah be pleased with him) who said: “The Prophet (peace and blessings of Allaah be upon him) used to relieve himself, then another boy and I would bring a vessel of water and a short spear [to use as a sutrah for the prayer he was going to do after doing wudoo’] and he would cleanse himself with the water.”
(Narrated by al-Bukhaari, 149; Muslim, 271)
The reason for that is that the basic way of removing impurities is to use water. Just as you use water to remove any impurities from your foot, so too you use it to remove any impurities resulting from relieving yourself.
2 – He may purify himself using stones. Doing istijmaar or removing impurities with stones is sufficient. This is indicated by the words and actions of the Messenger (peace and blessings of Allaah be upon him). With regard to his words, Salmaan (may Allaah be pleased with him) said: “The Messenger of Allaah (peace and blessings of Allaah be upon him) forbade us to use less than three stones for the purpose of istijmaar.”
(Narrated by Muslim, 262).
With regard to his actions, Ibn Mas’ood (may Allaah be pleased with him) said: “The Prophet (peace and blessings of Allaah be upon him) went out to answer the call of nature and asked me to bring him three stones. I found two stones and searched for a third but could not find one. So I took a piece of dried dung and brought it to him. He took the two stones and threw away the dung and said, ‘This is a filthy thing.’”
(Narrated by al-Bukhaari, 122).
The hadeeth of Abu Hurayrah, which states that he gathered some stones for the Prophet (peace and blessings of Allaah be upon him) and brought them to him in his garment, put them down next to him then went away (narrated by al-Bukhaari, 154) indicates that istijmaar (using stones to cleanse oneself after relieving oneself) is permissible.
3 – Cleaning oneself with stones then with water.
I do not know of any report to this effect from the Prophet (peace and blessings of Allaah be upon him), but there is no doubt that this is more effective in cleaning.
Al-Sharh al-Mumti’, 1/103-105
Based on this, you have no excuse for not praying or for delaying the prayer beyond the appointed time because you are not able to do instinja’ (wash with water after relieving yourself), because you can remove the impurity and cleanse yourself of it using tissues and the like. Everyone can carry some tissues with him in his pocket to clean himself with. Moreover, it is not clear from the question what is stopping you from cleaning yourself with water after relieving yourself, especially since you say, “Should I make wudoo’ and pray or miss the prayer?” What that means is that water is available, so you can bring some water to the toilet and use it to clean yourself. If you do not do that then you have to remove the impurity using tissues and the like, then do wudoo’ and pray. It is not permissible for you to delay the prayer until the time for it has passed.
And Allaah knows best.