AOA brothers and sisters! i have a question that i got to know from one of my friends. Here is it

A guy and a girl who are above 25 both are muslims. They both love each other too much but they got to find out this thing that this relation of theres meeting and talking that is not legal in ISLAM. They stoped them selfs too much but they couldnt help there feelings and love for each other. So they decided to do secret nikah so that they can meet and talk and safe them selfs from doing any more sins. Basically they wanted to change there Haram relation to Halal. So they went to do nikah in a mosque with two witnesses. One witness was from the guy side and one witness was from the girls side. They told every thing to the Imam of the mosque. The Imam said that this is a very nice thing that u realized that you are doing wrong and you want to make it right. It wont be a sin to do a nikah in secret cause you are doing this to safe your selfs from SINS. Than the imam said that i am not a registered Imam cause this is a private mosque. All the nikah that have to be done are registered in GOVT mosques. But i can tell u this that you can perfrom nikah your self also. The guy asked the Imam, that why dont you perform it in secret. He said i cant cause if some time you have to told that you did nikah your self, then the law will capture me.
The Imam than told the guy how to perform nikah him self. Imam told the guy that take a paper and write all the things that is written on Nikah Nama. The Imam told the guy and the guy wrote every thing on paper. Than the imam said that now the girl should select a person to be his wali. The girls requested one of the witness to be his wali. The wintness accepted to be his wali. Than the guy asked both the witness that with your permission i wana do nikah with this girl do u allow this. Both the witness said yes we allow it. Than the guy started to say to the girl that i wana do nikah with you and this much haqmahar do you agree. The guy said this thing that have to be said three times and the girl said a yes three times. Than they all signed the Nikah Nama that they made by them selves.

I just wana know that is this Nikah legal according to our religion. Are they both husband and wife? can they satisfy each other sexually now?

I know that this Nikah is not legal according to the law cause there nikah didnt got registered. Secondly i have heard people doing these types of Nikah. So i wana ask u people, maybe one of you know better about this. So that i can guide my friend properly.