It's not a Lada.
Poland was producing a licensed version of the Italian FIAT 125 for years, it was sold all over the world, including UK, Cuba and China under the name Polski (or more precisely Polski Fiat), then renamed to FSO, as the license expired in '83. They were produced in Warsaw from '67 to '91.
It had the body of the 125 Italian model, but the 1300/1500 ccm engines of the older 124. Ladas were licensed 124's and modernised by the Ruskis.
The Polish ones were almost always marked with a p at the end of the model name: 125p, 126p, 127p, 130/132p. Most of them were 125 and 126, they were very popular in Poland.
The square headlights were not installed in Poland, it is more likely an Italian version. Of course there were some exceptions (in the beginning of the production for example).
Which engine does it have?
I'll try to help, but could be tricky, as the parts will differ between Polish and Italian factories.
On the first photo you have the 'Italian' square headlights front with a Fiat badge and next to the right the 'Polish' round headlights with a Polski Fiat badge.
On the second photo a rally version.
On the third photo a 125p pickup + a piece of the 127p
On the fourth photo a prototype, a 3 seat wide body pickup
The photos were taken at en exhibition in the former factory 3,5 years ago. It was held by individual owners of the cars
Quite impressing.
The last attachment is the google satellite view of the FSO factory terrain by the Vistula river in Warsaw. You can see the remains of the test track between the river and the factory.