The Republicans will try to install a new judge for the Supreme Court before the election.
Can the Democrats prolong this process, e.g. by filibuster, so that it does not finish on time?
It is not possible for Democrats to filibuster the nomination under the current Senate rules, due to Mitch McConnell's use of the 'nuclear option' in 2017 which allowed a nomination debate to be ended by a simple majority vote.
In 2013, the Senate voted 52-48 to change the number of votes needed for a successful cloture vote (a vote to end a filibuster) on all nomination debates except those to the Supreme Court from 60 to a simple majority vote. This step was taken by the Democrats due to continued filibustering of appointments to lower courts by the Republicans, and was opposed by all Republican senators and three Democratic senators.
This measure was then extended to apply to Supreme Court nominations in 2017, with the Senate again voting 52-48, this time splitting along party lines; Republicans in favour, Democrats against. This was in order to prevent Democrats filibustering to block the appointment of Justice Neil Gorsuch to the Supreme Court.
As a result, a filibuster can be ended by a cloture vote supported by a simple majority of senators (or, in the case of a tie, the Vice President's tie-breaking vote). As there are currently 53 Republican senators, the party can end any filibuster attempt. The timing of the vote is also in the hands of Senate Majority Leader McConnell.
The rest of the nomination timetable is also in the hands of the Republican party; the nominee has to first be confirmed by the Judiciary Committee, chaired by Republican Lindsey Graham, and with a Republican majority.
As a result, there is no procedural way for Democrats alone to block or delay a nomination so that it does not finish on time. Bear in mind also, that there is currently nothing to stop the Senate confirming the nomination within the lame duck period between the November election and the convening of the Senate at the beginning of January.