The EPL will not wait until October, two full months after finishing this season, so that a small group of teams can play out their remaining UEFA matches. These are typically played concurrently anyway, so why should it be any different?
I'll grant you the start of the qualifying rounds for the 2020-21 European competitions may need to be pushed back for the current domestic seasons to be completed (qualifiers normally would have begun on June 23!), but this won't affect domestic schedules except for four clubs Europe wide (the finalists). All matches except the final are played midweek anyway (and even that could change for this one season).
Then the group stage matches for the two Euro competitions will probably have a slightly later start date, but it should all be caught up by the end of the calendar year. Fixture lists will be crowded for sure, but taking a two month break after finishing these final nine matches wouldn't make any sense at all.
The European games will be played at midweeks and weekends. There's so many games left that they have to play at every opportunity across the full month. Obviously if they are not played, then the EPL season can start earlier than October.
What do you think the last 10 weeks have been? It's probably the longest break footballers have had in their entire career.
During the break, the fixture list has to be arranged, clubs have to relay pitches, promoted clubs have to improve their grounds, we're moving to a new training facility, players are leaving clubs and joining new ones. There's European tournaments in August to complicate it all, and major international tournaments at the end of next season.
I know that players have just had a long lay off which is now causing a spike in injuries, but people expecting them to play every week for the next 12 months or more, with just a 1-2 week break is not realistic. So as things stand, I can't see the new EPL season starting before October at the earliest.