That got me thinking... seems I'd noticed that before.
Looking at IMFDB, they have it as a 'Steyr Mannlicher-Schoenauer'. They also show what appears to be a GK or Model 1952 by way of example (White Heat was released in '49): White Heat IMFDB

Of the three stills shown on IMFDB, one shows the right side though not very well. It seems that the bolt handle is rather close to the trigger guard, unlike an MS:

Then I watched the following clip. At 00:06 - 00:07 the right side is plainly visible and the bolt clearly closes aft of the rear receiver bridge, to which a scope ring is attached. Notice also how he works the bolt at 00:33. It is not a Mannlicher Schoenauer.

White Heat: White Heat Clip