CONSIDERATION BY THE SENATE
As mentioned by the CRS Report, only the Senate procedures are mentioned by the law creating the Joint Select Committee on Budget and Appropriations Process Reform. Nothing is mentioned about the House. The CRS follows the language from the statute, and it helps in understanding how the procedures work:
What happens once the committee reports?
If the committee approves the report and legislative language, it must make them available to the public “promptly” and submit them to the President, the Vice President, the Speaker of the House, and the majority and minority leaders of each chamber within 15 calendar days of approval.
Upon receipt of proposed legislative language, the Senate majority leader (or his designee) shall introduce it in the Senate (by request) on the next day on which the Senate is in session. There are no provisions in the law concerning the introduction of the recommendations of the joint select committee in the House.
What are the procedures for congressional consideration of the committee’s recommendations?
The Bipartisan Budget Act [of 2018] establishes certain unique procedures for Senate consideration of any legislative language reported by the joint select committee. These procedures are intended to allow the Senate to reach a timely vote on the question of whether or not to consider legislation embodying the recommendations of the joint select committee, but the act does not specify any procedures governing consideration of the bill once the Senate has agreed to take it up. There are no provisions in the act concerning the consideration of the recommendations of the joint select committee in the House. There are also no provisions concerning resolving any differences between the House and Senate or the consideration of a veto message from the President. Such actions would occur under the regular procedures of each chamber. Furthermore, the act includes language recognizing the constitutional authority of the Senate to change rules relating to procedure in the Senate at any time, in the same manner, and to the same extent as in the case of any other rule of the Senate. 
Committee Consideration in the Senate
Once any recommendations of the joint select committee are introduced in the Senate, the bill is to be referred to the Senate Committee on the Budget, which shall report the bill favorably, unfavorably, or without recommendation within seven session days—but without any revisions. If the Budget Committee fails to report the bill within that period, it shall be automatically discharged from consideration of the bill, and the bill shall be placed on the Senate Calendar of Business. 
The Motion to Proceed in the Senate
Not later than two days of Senate session after a joint committee bill has been reported or discharged from the Budget Committee, the majority leader (or his designee) may move to proceed to consider it. Should the majority leader (or his designee) not make such a motion within two session days, any Senator may do so. 
The motion to consider a joint committee bill—and all debatable motions and appeals in connection with the motion—shall be considered for a maximum of 10 hours, evenly divided between the majority leader and the minority leader (or their designees). A non-debatable motion to further limit debate would be in order and would require a vote of three-fifths of all Senators – 60 votes if there is not more than one vacancy—to pass. 
In order for the recommendations of the joint select committee to be considered by the full Senate, the act requires that the motion to proceed be agreed to by a vote of three-fifths of all Senators—60 votes if there is not more than one vacancy. 
The act further specifies that all points of order against the motion to proceed are waived and that a motion to postpone the motion to proceed or a motion to reconsider a vote on it are not in order. 
Finally, the act directs that not later than the last day of the 115th Congress (2017-2018), the Senate must vote on a motion to proceed to a bill containing recommendations of the joint select committee. 
Floor consideration in the Senate
If the Senate approves the motion to proceed, the joint committee bill may be considered under the regular rules of the Senate so that it will be fully debatable and fully amendable (possibly including by non-germane amendments). Cloture may need to be invoked on one or more questions (requiring the support of three-fifths of all Senators) in order to reach a final vote on it.