Why is it runway "1" instead of "01" in America? In the case of a runway with a magnetic heading of 10° in America the runway number ("1") isn't hugely distinguishable from the runway centreline IMHO, (for example see Washington Airport and San Francisco Airport). 

Is there a reason why the USA doesn't add the leading zero like the EU (example Brussels Airport and RAF Waddington)?

Does the ICAO make any recommendations about this?
Indeed USA deviates from ICAO:

5.2.2.4 Zeros are not used to precede single-digit runway markings. An optional configuration of the numeral 1 is available to designate a runway 1 and to prevent confusion with the runway centerline.

Source: USA AIP - GEN 1.7 Differences From ICAO SARPs
5.2.2.4 in ICAO SARPs Annex 14 (Aerodromes) is as follows (emphasis mine):

5.2.2.4 A runway designation marking shall consist of a two-digit number and on parallel runways shall be supplemented with a letter. On a single runway, dual parallel runways and triple parallel runways the two-digit number shall be the whole number nearest the one-tenth of the magnetic North when viewed from the direction of approach. On four or more parallel runways, one set of adjacent runways shall be numbered to the nearest one-tenth magnetic azimuth and the other set of adjacent runways numbered to the next nearest one-tenth of the magnetic azimuth. When the above rule would give a single digit number, it shall be preceded by a zero.

Shall in ICAO legalese denotes a standard.
The aforementioned optional configuration for &quot;1&quot; covers your concern regarding the centerline:


The numeral &quot;1&quot;, when used alone, contains a horizontal stroke, as shown, to differentiate it from the runway centerline marking.

Source: FAA AC 150/5340-1
The rationale is not in the AIP; it's not always given. Deviations from SARPs are also very common, that's why there's an AIP section dedicated to them.