The UK could, in theory, claim their entire EEZ in case of a no-deal Brexit. (At a substantial cost in other areas of the economy.)
If we left the EU it wouldn’t necessarily mean that the situation would improve. The House of Commons Library has said that “many of the underlying issues that affect fisheries management would remain unchanged.”
The UK is signed up to the UN Law of the Sea Convention which allows countries to establish an Exclusive Economic Zone of up to 200 nautical miles from their coast. If the UK were to leave the EU we could have control of all fish which were within this zone. But, the same laws also require countries to ensure that fish stocks are conserved and that the allowable catch is specified and where necessary shared with other countries.
Countries such as Norway have this 200 nautical mile Exclusive Economic Zone in place, they also have agreements with a number of countries, and with the EU to allow fishing in those waters.
Of course, doing any other (trade) deal with the EU will probably involve relinquishing some of this exclusivity; see Norway.
The EU current system is limited to 100 miles of such exclusive claims:
The European Union’s Council of Ministers sets the tonnage of specific fish, or ‘Total Allowable Catch’ (TAC), which can be caught within EU waters and then divides this between each member state.
The quota each member state receives is based largely on how much they fished in those areas in the 1970s, before the Common Fisheries Policy came into effect.
This method of splitting up the stocks is designed to keep the levels of fishing in these areas relatively stable. However, it has been criticised in the UK, particularly as for much of the 1970s the UK fishing fleet spent much of its time in the waters around Iceland, which is not a member of the EU.
In 1976 the UK fleet was expelled from this area when fishing limits around Iceland were extended to 200 miles.
Other methods of dividing fish stocks have been proposed, but the House of Commons Library has said the system is unlikely to change.
It’s also impossible to determine what the policy on fishing would be if we left the EU and things wouldn’t necessarily be any better.
Currently EU member states are allowed to place limits on who can fish in their territorial waters, and up to 100 nautical miles fishing is restricted to those who traditionally fished there, but the legislation covering this expires in 2022. Whether or not this will be replaced by that date will be a matter for the politicians to determine.
Also unilateral setting of quotas are likely to lead to fishing wars:
The transnational nature of fisheries may limit the UK’s options after leaving the CFP.
Setting unilateral quotas would likely cause international disputes, such as the ‘mackerel wars’ of 2010-11, when negotiations between the European Commission and Iceland and the Faroe Islands broke down.
It is likely that the UK would have to continue negotiating its total allowable catch after Brexit, with it signed up to international agreements that require it to ensure “proper conservation” of fish stocks and to co-operate with regional or global organisations in achieving this, especially where stocks are shared.
The UK’s ability to restrict access to its waters may be limited by historical claims by European fishermen. In the past, tribunals and international courts have often ruled in favour of historic access rights to waters and against those looking to limit access.
In the mackerel wars, the EU embargoed mackerel exports from those islands in response to what they perceived to be excessive quotas.
As of 2017 the mackerel conflict was not really resolved with respect to Iceland:
A few years after the shift in mackerel distribution, the
Coastal States attempted to include Iceland into their negotiations.
Eventually, Iceland was granted [NEAFC] “observer status” in
2008 and Coastal State status in 2010. Nonetheless, until the
time of writing, Iceland has not been involved in the Coastal
States’ agreements on the total allowable catch (TAC) and
quota allocations per country. The main reason for this failure
is that a social and political dispute between the Coastal States
developed which persists to this day. The conflict prevents
collaboration in a joint management plan and subsequently
sustainable management of the stock
On the other hand, the Faroe Islands did strike a deal that increased their mackerel quotas:
The parties have recently settled their dispute in respect of mackerel. On 12 March 2014, the Faroe Islands, Norway and EU concluded a joint arrangement for the conservation and management of the North East Atlantic mackerel stock for the next five years. The arrangement allocated 13% of the TAC between the parties (not including Russia and Iceland) to the Faroe Islands. This is a sizeable increase compared with the 5% that had been previously allocated to the Faroese, and the proportion is set to increase again next year.
The Faroe Islands had another ongoing dispute with the EU on herring at that time (2014), so the EU did not lift their sanctions right away. In both cases (mackerel and herring) Denmark (who represents the islands) took the case to the WTO and to the PCA.
Pending an agreement on herring, the WTO complaint and the UNCLOS Annex VII arbitration continue, and EU Regulation 793/2013, establishing sanctions against the Faroe Islands, remains in force.
According to the EU document that did lift the sanctions on the Faroe Islands later that year, the latter gave up on their herring claims.
Basically even without EU membership, the UK has to negotiate such quotas with other countries, e.g. in the North-East Atlantic Fisheries Commission (NEAFC).
Fish stocks that reside in the international
waters of the northeast Atlantic, like the mackerel, are
multi-laterally managed by the North-East Atlantic Fisheries
Commission (NEAFC), where the contracting parties—the
EU, Norway, Iceland, Russia, and Denmark (on behalf of
the Faroe Islands and Greenland)—negotiate the division of
fishing resources. When stocks frequent the national waters of
a nation, defined as a 200-mile exclusive economic zone
(EEZ), that nation is considered a “Coastal State” for that
stock and has the right to harvest it in their EEZ.
The management of the straddling stocks in the northeast
Atlantic is a two-tier process: Coastal States agree on shares
and management plans before bringing the matter to NEAFC
to cover fisheries in waters outside national jurisdiction
(Vanderzwaag and Russell 2010). The regulatory area of
NEAFC consists of the North-East Atlantic Ocean, the
Barents and Norwegian Seas, and the Arctic Ocean (Fig. 1).
So whether the UK will be better off (in fisheries) comes down to the more general issue "we'll strike better (trade, fisheries etc.) deals on our own after Brexit."
Whether the UK can exploit all its fishing quotas by itself or grants them to pseudo-UK companies that are foreign owned is a somewhat different matter. But the benefit from the UK granting such quotas for foreign companies is not zero because of the fees, levies and royalties they could impose. (See Australia or Canada for example(s).) Of course the benefit for the national economy is greater if all the industry is nationally owned. Also, there's the issue of workforce nationality, which the UK could regulate in its favor even when in grants the quotas to foreign-owned companies. Supposedly 92% of the British fishermen voted for Brexit. Whether they did this because they think deals will be struck or fishing wars are easy to win... I don't know.
Iceland supposedly abandoned its EU membership application in part due to the mackerel war. Of course, for a country where the fishing industry is (now) central to their economy after the failure of their internationally-oriented financial sector... this is a little easier to comprehend.
During the 1976 cod war that Iceland mostly won, they had threatened to leave NATO. And they are talking again of quitting NATO, although this latter occurrence seems to be along the now entrenched lines on this matter in Icelandic politics.
Not surprising, the Conservative-leaning Telegraph has an article titled "After Iceland won the cod wars, their independent fisheries thrived. Britain can do the same"; it's penned by Owen Paterson, whom Wikipedia describes as "a leading supporter of Brexit".
The catch with a EU-UK fishing war is that
The UK exports 45% of its catch. 80% of that quantity goes to EU countries. For example 90% of fish landed in Ramsgate are sold in the Boulogne Fish Market - for 15% more in value than they would get at home.
In case of a fishing war (and a probable EU embargo), they'd have to find new markets.