You will need working visas for most South American countries. For the major South american countries:

1. Brazil: Bring to the Brazilian embassy your passport valid for at least six more months, negative criminal records (at most 3 months old), two copies of a 5cmx7cm photo (passport), some filled forms available at the embassy. All documents must be in Portuguese (new Ortographic Treaty), if any must be translated, it must be done by a professional translator. A temporary work visa is valid for 2 years. Official Info

2. Argentina: UK nationals do not require a visa: http://www.migraciones.gov.ar/accesible/indexN.php?visas
   2.1 As noted in the comments by CMaster, it is better to contact the embassy. The website may be outdated.

3. Chile: You need to request a special working permit: http://www.extranjeria.gob.cl/ingles/permiso_visa_tra.html

4. Colombia: You need to get a TP15 temporary visa:

TP 13: For the foreigner who wishes to enter Colombian territory to provide technical assistance in his/her area of expertise, with or without, work contract to public and private entities.

It is also worth of note that you won't be "working" in the traditional sense (being employed by a national company or a local subsidiary AND being paid while in the country), so some regulations may not apply.

You will need working visas for most South American countries. For the major South american countries:

1. Brazil: Bring to the Brazilian embassy your passport valid for at least six more months, negative criminal records (at most 3 months old), two copies of a 5cmx7cm photo (passport), some filled forms available at the embassy. All documents must be in Portuguese (new Ortographic Treaty), if any must be translated, it must be done by a professional translator. A temporary work visa is valid for 2 years. Official Info

2. Argentina: As noted in the comments by CMaster, contact the embassy. The website may be outdated.

3. Chile: You need to request a special working permit: http://www.extranjeria.gob.cl/ingles/permiso_visa_tra.html

4. Colombia: You need to get a TP15 temporary visa:

TP 13: For the foreigner who wishes to enter Colombian territory to provide technical assistance in his/her area of expertise, with or without, work contract to public and private entities.

It is also worth of note that you won't be "working" in the traditional sense (being employed by a national company or a local subsidiary AND being paid while in the country), so some regulations may not apply.

Also since OP is self-employed, he won't be working for any nationals and probably won't be receiving any money during his travel (even though his company may), maybe it does not fall into a work visa. Contacting the embassy of the destination may clarify if he indeed needs a work visa or not.

You will need working visas for most South American countries. For the major South american countries:

1. Brazil: Bring to the Brazilian embassy your passport valid for at least six more months, negative criminal records (at most 3 months old), two copies of a 5cmx7cm photo (passport), some filled forms available at the embassy. All documents must be in Portuguese (new Ortographic Treaty), if any must be translated, it must be done by a professional translator. A temporary work visa is valid for 2 years. Official Info

2. Argentina: UK nationals do not require a visa: http://www.migraciones.gov.ar/accesible/indexN.php?visas

3. Chile: You need to request a special working permit: http://www.extranjeria.gob.cl/ingles/permiso_visa_tra.html

4. Colombia: You need to get a TP15 temporary visa:

TP 13: For the foreigner who wishes to enter Colombian territory to provide technical assistance in his/her area of expertise, with or without, work contract to public and private entities.

You will need working visas for most South American countries. For the major South american countries:

1. Brazil: Bring to the Brazilian embassy your passport valid for at least six more months, negative criminal records (at most 3 months old), two copies of a 5cmx7cm photo (passport), some filled forms available at the embassy. All documents must be in Portuguese (new Ortographic Treaty), if any must be translated, it must be done by a professional translator. A temporary work visa is valid for 2 years. Official Info

2. Argentina: UK nationals do not require a visa: http://www.migraciones.gov.ar/accesible/indexN.php?visas
   2.1 As noted in the comments by CMaster, it is better to contact the embassy. The website may be outdated.

3. Chile: You need to request a special working permit: http://www.extranjeria.gob.cl/ingles/permiso_visa_tra.html

4. Colombia: You need to get a TP15 temporary visa:

TP 13: For the foreigner who wishes to enter Colombian territory to provide technical assistance in his/her area of expertise, with or without, work contract to public and private entities.

It is also worth of note that you won't be "working" in the traditional sense (being employed by a national company or a local subsidiary AND being paid while in the country), so some regulations may not apply.