The French are known for having a more independent foreign policy, and thus more leeway when it comes to decision making, but I would give this one to the UK. On a tit-for-tat basis:
Submarines:
The French have 6 Rubis-class nuclear attack submarines, which are the most compact nuclear submarines created, displacing some 2,660 tonnes. The UK has 7 Astute-class nuclear attack submarines, actively replacing the 7 Trafalgar-class nuclear attack subs. The Trafalgar-class displace 5,300 tonnes and the newer Astute-class displace 7,400 tonnes each. The Astute are widely regarded as a rival to the US Virginia-class, being the best or second best attack subs in the world in terms of advanced equipment. The older Trafalgar-class are exponentially superior in every way to the Rubis-class, so I'd give this one hands down to the UK. The French are replacing their Rubis with the 5,300 ton Barracuda-class, but it will be ~5 years until the first commissioning, and I suspect they will still be inferior to anything the UK has been churning out.
Both navies operate 4 nuclear-powered ballistic submarines, the UK the 15,900 tonne Vanguard-class, and the French the 14,335 tonne Triomphant-class. Obviously, most details of the sensor suite and processing systems are usually either classified or watered down for the public, so we can't make a one-for-one comparison. The Vanguard-class has impressed the US Navy, and was said by some to be superior to the Ohio-class sub so I'd give this one to the British as well.
Carriers:
The French have the only catapult-launched, nuclear-powered carrier outside of the United States. The Charles De Gaulle displaces 42,000 tonnes, and carries 28-40 aircraft, including the Super Étendard and Rafale combat aircraft. It has experienced countless problems and has had two retrofits already (with another scheduled for 2017). The UK does not have a carrier as of right now, but the Queen Elizabeth-class will produce two carriers (commissioning in 2016 and 2018), both nuclear-powered and displacing 70,600 tonnes (though they will likely displace more over their lifetime). They will carry 40-50 aircraft, including the F-35B. It will be operated as a STOVL carrier, which reduces the range and payload capability of the aircraft (no catapult-assisted). I would still give this to the British, for having two carriers, at double the displacement, operating more aircraft (that are more advanced in sensors and avionics), and most probably having significantly less maintenance issues. But, as of right now, they don't exist, so the French have a superior carrier force for now.
Amphibious warfare:
The French currently have three of the 21,000 tonne Mistral-class, while the UK operates one 21,500 tonne Ocean-class and two 19,560 tonne Albion-class warships. While the Mistral is superior to the Albion in almost every way, it is more evenly matched when it comes to the Ocean. They both have approx. the same vehicle capacity, troop capacity, and aircraft carrying capacity. The significant difference is in the superior armament the Ocean-class carriers (although all in all it doesn't make much of a difference when what you're carrying are just machine guns). There is ONE Ocean class while there may be up to SEVEN Mistral class put into commission. France has the superior amphibious assault force.
Destroyers and Frigates:
The French operate 10 large surface ships while the UK operates 19. The British Type-45 class is one of the most technologically advanced destroyers in the world. The largest French destroyer is the 7,050 tonne Horizon-class (the French have two), which is slightly smaller than the 8,500 tonne Type-45 (of which the British have six). It's the same for the smaller destroyers and frigates: The British generally operate more, newer ships, with superior quality control and firepower. Comparing the Type-45 to the Horizon, the Type-45 is faster, has longer range, a larger complement, and has a superior sensor suite. The anti-air system is largely the same, both equipping 1 × 48-cell
Sylver A50 VLS, for a combination of 48:
Aster 15 missiles (range 1.7-30 km) and/or
Aster 30 missiles (range 3-120 km). In terms of anti-ship missiles, the Horizon operates 8x Exocet and the Type-45 has 2 x quad Harpoons (for a total of 8). The Type-45 relies on their two ASW helis while the Horizon relies on their 2x MU90 Impact double torpedo tubes and 2x SLAT anti torpedo system. The Type-45 has the lead in a tit-for-tat show, but at best they are evenly matched. The UK operates six and the French operate two so the lead definitely goes to the UK. This is largely the same at the level of the smaller destroyers and frigates. The UK is slightly qualitatively superior, but significantly quantitatively superior.
Conclusion: In terms of equipment, the UK definitely takes the lead. This obviously excludes battle doctrine, training, and experience, but I'm sure that these are largely the same in both countries. It will come down to who has better equipment, and at least for the Navy, the UK leads significantly.
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