Why is 英语和汉语我都会说 correct and 我会说都英语和汉语 incorrect?
Yes, without 都, you can say "我会说英语和汉语" The reason is unlike the English word "both" which is placed before the objects,  "都" must be placed after the objects


I can speak [English and Chinese]
I can speak [both] [English and Chinese]
我会说[英语和汉语]
[英语和汉语]我[都]会说 
我[英语和汉语][都]会说 


Notice in #4 and #5 the object [英语和汉语] is placed before 都

"我会[都]说 [英语和汉语]" and "我会说[都][英语和汉语]" violated this rule.

Note:

"都" in "我都会说英语和汉语" (I too, can speak English and Chinese) means "also" or "too". It is different usage of "都"