Do you know how much this registration usually take?
I do not.
Will I have any problems if my documents will be ready only in several months?
The freedom of movement directive, 2004/38/EC, has this to say, in Article 5, paragraph 4 (edited to remove clauses that do not apply to you):
- Where a Union citizen ... does not have the necessary travel documents ..., the Member State concerned shall, before turning them back, give such persons every reasonable opportunity ... to corroborate or prove by other means that they are covered by the right of free movement and residence.
Now that concerns the right of entry; the right of residence can be conditioned on your having a valid passport or identity card, and the right of residence for more than three months can be subject to registration, but the details are left to national law. In general, though, your status as an EU citizen should not depend on your possession of any given document; a document serves only as proof of your citizenship, from which the rights you enjoy flow directly.
If you have copies of the evidence you submitted to demonstrate your Hungarian nationality to the Hungarian authorities, and/or receipts for your pending applications, they should suffice along with your foreign passport to allow you to remain in the Schengen area and to travel to Hungary. You're more likely to run into trouble if you try to leave and re-enter the Schengen area, however, so if I were you I would avoid that if possible.
Will I be able to take the train from Paris to Budapest to apply for my passport?
Yes, but bring all your supporting evidence in case you are inspected by any border officers.