The UK parliament has already passed on Dec 20 the 2nd reading of the Act approving the exit deal... and this time (unlike before the latest UK general election) the new Parliament also passed an accelerated timetable for the final adoption of this bill:
Now that the bill has passed, lawmakers will have another three days to discuss it further, beginning on January 7. Lawmakers could possibly make amendments to the bill. The final vote will take place on January 9.
As far the exit agreement alone is concerned, there's rather limited scope for further EU Parliament or other national parliaments involvement:
The EU parliament, including the MEPs from the UK, must consent by simple majority to the Withdrawal Agreement – but does not have the power to amend it. [...] But there is no role for national parliaments of the 27 member states in the context of the Withdrawal Agreement, meaning for example that the French, Spanish or Polish parliaments do not have to agree to it.
On other hand, when it comes to future relationship, there are potentially a lot more hurdles...
When it comes to any future trade agreements, it is unclear whether the UK parliament will get the enhanced involvement in ratifying it that it has in the context of the Withdrawal Agreement, or whether only the ordinary rules relating to treaties apply, as the government has implied. The House of Commons Committee on Exiting the European Union has been critical of this, calling the ordinary process “inadequate” as it does not guarantee “a debate or vote on a treaty before it is ratified.” [Note that this was the old, pre-election UK Parliament saying this.]
However, the situation on the EU side is constitutionally more complex than with the divorce part of the Brexit process. In addition to the EU parliament, which must consent to the future trade agreement, all 27 member states will need to ratify it according to their national constitutional provisions.
Possibly in line with a Johnson election promise to the Brexit Party [voters] but surely also enshrined in the Conservative party election manifesto, this [revised] Act included a change (with respect what was proposed in the similar Act before the general election) that eliminated any extension to the transition period.
Johnson’s move to make his manifesto promise legally binding was mooted during the election campaign. His pledge not to ask for another extension was used by the Brexit party leader, Nigel Farage, as the reason why he would stand down his candidates in 317 Tory-held seats.
Now some of the EU trade negotiators, including Irishman Phil Hogan, who is the new EU Trade Commissioner, have already said this deadline is unrealistic... and Hogan also affirmed his belief that Johnson will probably break this pledge rather than have no deal with the EU on future trade.
"I don’t believe Prime Minister Johnson will die in the ditch over the timeline for the future relationship either ..."
Bloomberg has posted a fairly detailed tentative schedule of future steps. The interesting part of that is that on the EU side they'd have to decide whether to attempt a more comprehensive deal or go for something simpler...
Jan. 29-Feb. 1: The European Commission in Brussels will seek a mandate from EU governments for negotiations on a free-trade agreement with Britain. The key question, still being weighed within the commission, is whether this proposal should also cover the other areas such as fisheries, aviation and security that will form part of the future EU-U.K. relationship. A single mandate for the whole package would facilitate a green light by EU governments while potentially complicating any subsequent desire by the bloc to reach a quicker, narrower, deal in 2020.
Interestingly, depending on the scope for the negotiation mandate, it could require either unanimity in the EU Council or just qualified majority. However given the likely broad scope here unanimity is more likely to be required (see pp. 57-58 in https://www.asser.nl/media/4140/cleer17-5_web.pdf for a detailed discussion.)
After this, progress on negotiations will be formally assessed by EU leaders in some EU summits in March and then in June, with June 30 being the "Deadline for deciding whether to prolong the transition period by one year or two years" (which of course Johnson opposes and has even put into UK law.)