I don't think anyone would make that connection between Frank and Pat Butcher unless you pointed it out, as your DSs wouldn't be a couple. I could only see family doing it in a fond way.
In my experience, lots of people don't even know Jamie Redknapp and Frank Lampard are cousins so I wouldn't worry unless your family are particular football fanatics.
If someone I knew, however, called their son Jamie Oliver I probably would have a chuckle, particularly since he's likely to be famous as your DS grows up. To me it's a bit like in Gavin and Stacey where the baby is called Noel Edmund and everyone laughs.
I think if we go with Patrick we'd call him Paddy but I guess Pat might happen naturally.
I'm def over thinking - problem is I still have 6 months to think bangbang that Gavin and Stacy scene is hilarious...not sure we want to recreate it. Although he'd be introduced as James Oliver so hope people wouldn't link to chef? We'd just call him Jamie as a nick name. Oliver is only a middle so wouldn't be used much anyway.
Could use John as a middle which gets around it as its another family name.
Unless your surname is Butcher I don't think you need to worry about Frank and Pat references - would never ever have occurred to me, although I've never watched Eastenders so only have a vague knowledge of who they are.
Jamie Oliver, hmm personally I wouldn't use as I think everyone would think of the chef, sorry. Although I had no idea there was any link between Jamie and Frank - you're definitely over-thinking that one and I'm sure if you pick almost any two names most will have some sort of vague connection.
To be honest once he gets past the baby stage where you're introducing him to lots of people he probably won't be known as his full name (eg: Jamie Oliver F....) or be being introduced to people with his Dad (eg: Frank and Pat) that often anyway and so long term I don't think either really matter.
The pat and frank and the Jamie and frank thing you are way over thinking! (Unless your surname happened to be Butcher) The Jamie Oliver thing could be an issue but then again how often would you use his full name? I can't think of many people who know my dd's middle name. Do you like James? Call him James Oliver officially then use Jamie as a nickname?