It was narrated from the Prophet (peace and blessings of Allaah be upon him) that it is haraam to pray in certain places, including bathrooms. It is not permissible to pray in them. It was narrated in a saheeh report from the Prophet (peace and blessings of Allaah be upon him) that he said: “All the earth is a mosque except for graveyards and bathrooms.” (Narrated by al-Tirmidhi, al-Salaah, 291; classed as saheeh by al-Albaani in Saheeh Sunan al-Tirmidhi, 262). “Bathroom” refers to every place that is used for the purpose of relieving oneself; it is not permissible to pray in any such place. This room on the edge of which there is a toilet is not considered to be a bathroom so long as there is a barrier between it and the bathroom, with a wall and door etc. In this case prayer in that room is valid, because the prohibition on praying in bathrooms is because they are usually naajis (impure), and this room is not like that. And Allaah knows best.
What indicates that prayer in this place and elsewhere is valid is the hadeeth in which the Prophet (peace and blessings of Allaah be upon him) said: “The earth has been made a place of prayer and a means of purification for me, so wherever a man is when the time for prayer comes, let him pray.” (Narrated by al-Bukhaari, al-Tayammum, 323). So prayer in that place is valid, and it is also valid in parks and parking lots, on condition that the place in which one prays is taahir (pure, clean).
Shaykh Ibn Baaz (may Allaah have mercy on him) was asked, if there is a bathroom in front of a person in the direction of the qiblah, is it o.k. to pray facing it? He said: The prayer is valid and there is nothing wrong with that, rather what is forbidden is to pray inside the bathroom.
See Fataawa al-Shaykh Ibn Baaz, vol. 2, p. 196.
One thing to which attention must be paid is that a woman should uncover her face when praying, especially when she is among other women or her mahrams. But if she is praying in a place where men can see her, then she should cover her face whilst praying. See the answer to question no. 21803.
Secondly: you will be rewarded, in sha Allah, for your keenness to do the prayers on time. Undoubtedly the Muslim is obliged to do the prayers on time. And you have to make yourself feel that you are obeying Allaah when you are praying.
Your adherence to the truth and your establishing prayer and not feeling shy to worship openly may be the means of guiding some of the non-Muslims who see you to Islam, in which case you will have a reward equal to theirs in sha Allaah.
May Allaah help us and you to do all that is good.