Ireland? EU/Schengen party states? Asian and South American countries?
The UK is part of the Five Countries Conference group, which agreed to share information about visa applications and other immigration intelligence between themselves - the other members of this group are Canada, the US, New Zealand and Australia. It is highly likely that a ban in the UK would affect applications to these countries.
The UK also has access to and informs the Schengen Information System, so there is data sharing occurring there - however, it is noted that as the UK is not a full member it cannot issue SIS alerts but there is no information on whether UK refusals or immigration issues are available to SIS users. It is possible but not certain that a ban in the UK would affect Schengen applications of travel.
The UK and Ireland are part of the Common Travel Area, which means both countries closely cooperate and share data between themselves - a refusal or ban in either country can certainly affect applications to the other country, due to the common travel area movement rules. It is highly likely that a ban in the UK would affect applications or travel to Ireland.
Many countries will ask in their visa forms if visa or entries to a third country have ever been denied.
Even if there is no automatic data sharing it is a bad idea to lie on a visa form. If you tell the truth, the visa may be denied because of the bad travel history, or it may be granted if the rest of the application looks good. If you lie and that gets caught, the visa will be denied and there may be the next ban.