It's not impossible that the UK would pull away from the US. That wasnt the reaction in OTL, perhaps due to the fears of Soviet invasion of western Europe. Especially after the crushing of the Hungarian Uprising around the same time as Suez.
As other posters have said, with both France and the UK out of the formal NATO structures the organisation is potentially moribund. Although of course France still retained formal membership and its forces in Germany would have fought alongside other NATO units. Would the UK be in a similar position?
Removing USAF units from the UK would seriously damage NATOs air power. On the upside, for the UK, it might not have been in as much danger of destruction during the Cuban Missile Crisis. If that wasn't butterflied away by the geopolitical effects of UK "Gaulism".
Probably no British purchase of Polaris missiles but instead the prolonged use of V bombers, possibly with a new version of the Vulcan optimised for low level strike and even early stealth ECM.
Another interesting POD arising from UK disillusionment with the US could be earlier British participation in the EEC. And a greater emphasis on military cooperation, in procurement at least, with France and later Germany etc.. Maybe sharing the development of SSBN and missile technology.