Why do the British pronounce "Thames" as "tem"?

Question

What language does that pronunciation originally come from?

Answer 1

The river was known to the Romans as "Tamensis" or "Tamisia" a name of Celtic, possibly even pre-Celtic origin (See Picts and Beaker folk / Beaker Culture).

I tend to support the pre-Celtic theory myself. I don't think that attempts to link the name to  words in Celtic languages meaning "darkness" or "dark gray" are very convincing.

When the Germanic Jutes and Anglo-Saxons came to England c. 499 A.D., they altered the Latin  pronunciation of the name (Tamisia) to Temes (Taym-ĕss).

By Shakespeare's time the pronunciation had simplified a little further to just one syllable, taymz." A few centuries later, popular pronunciation in England had changed the pronunciation once again to "tĕmz."  

In the United States, the pronunciation "taymz" is still generally used. The Thames River in Connecticut is pronounced "taymz." However in England, the pronuciation is now ""tĕmz" and maybe even "tĕm" in some instances as you imply.

The th in the name has always been silent like the th in Thomas. It was written in there simply by scholars but never pronounced by anyone.

Answer 2

Could have been the Roman Soldiers, Romano-Britons or French (Normans) who coined the name.