These three can be used in the same circumstances and can function the same in a sentence. For example Kahve içelim mi? (Shall we drink coffee?) "Peki", "olur" and "tamam" are acceptable answers in this question. Morever, "peki" means alright or well. Olur means possible and it is a verb (olmak: to become). Tamam means okey and it is also used in Arabic.

PEK quite İYİ 1good, 2fine "pek iyi" literally means "pretty good" though it is nowadays less common compared to the former decades. its evolved form "peki" has been more common for several decades but it means "alright" OLMAK 1to be, 2to become, 3to happen,to take place, 4to occur, 5to fit, 6to be suitable, 7TO BE ACCEPTABLE, 8to go well, 9to be completed, to be successfully done |and some more ol + ur (present simple tense) + - (personal ending) >olur 1it does be (sounds weird in English) 2it becomes 3it happens, 4it occurs 5it fits 6it suits 7IT IS OKAY, IT IS ACCEPTABLE in other Turkic languages such as Kazakh, Kyrgyz, Uzbek, you can hear "olar", it's again the simple present conjugation of the "olmak" verb. In Turkish of Turkey, "olur" and "tamam" are the synonyms. both mean "okay" the new generations have also started to use "ok" and "okey" directly loaning from English

Only a nuance. 95% of the time, they are same. Tamam is OK. Peki is more like "Alright" and has a slightly feeling of yielding, giving up. Olur is more like confirmation. More commonly used when we are asked a suggestion or a question with "Yarin sinemaya gidelim, olur mu?" Yet, you can still reply this question with "Tamam" Personally, I don't recommend you to spend time on that small difference. Just assume they are same.