How should I proceed with marrying my girlfriend who converted to Islam after becoming pregnant with my child? Is there any rule that I have to wait for 3 months to ensure purity?
Bismihi ta'ala
All you need to do is recite the Nikah ceremony, and there is no need to wait for a three month period.
If you have a temporary contract between you and her, you must nullify that first, by freeing her for the remaining period. Then you do the permanent Islamic marriage contract with her.
And Allah knows best.