while his sisters, convinced that no country advice could be of any service, recommended an express to town for one of the most eminent physicians.
句子不完整，缺少主句或并列分句。
这只是一个从句，while 是从属连词或并列连词。句子主语是 his sisters，谓语动词是 recommended，an express 是宾语，to town for one of the most eminent physicians 是 express 的定语。
convinced that no country advice could be of any service，是形容词短语，分析为 his sisters 的“非限制性定语” 或 “原因状语”，均可。

不能说 any country advice could not，只能否定主语，使用 no country advice could... 又如：
No one could do it.（正）   Anyone could not do it. （误）

语法规定：不能使用 any...not 格式。