Concerning in flight proximity to other aircraft Rule 17 of the Air Navigation Order - "Rules for Avoiding Aerial Collissions" is what you want to read.
It doesnt specify any specific distances but I'm pretty sure that if the commander of either aircraft did feel that a risk of collission existed that they would file an AIRPROX.
In fact I'll paste a copy of the ANO's text below as its kind of in context under this thread, please excuse the lack of formatting. You can get your own copy in CAP393 from the CAA:
http://www.caa.co.uk/docs/33/CAP393.PDF
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Rules for avoiding aerial collisions
17 (1) General
(a) Notwithstanding that the flight is being made with air traffic control clearance it
shall remain the duty of the commander of an aircraft to take all possible
measures to ensure that his aircraft does not collide with any other aircraft.
(b) An aircraft shall not be flown in such proximity to other aircraft as to create a
danger of collision.
(c) Subject to sub-paragraph (g), aircraft shall not fly in formation unless the
commanders of the aircraft have agreed to do so.
(d) An aircraft which is obliged by these Rules to give way to another aircraft shall
avoid passing over or under the other aircraft, or crossing ahead of it, unless
passing well clear of it.
(e) Subject to sub-paragraph (g), an aircraft which has the right-of-way under this rule
shall maintain its course and speed.
(f) For the purposes of this rule a glider and a flying machine which is towing it shall
be considered to be a single aircraft under the command of the commander of
the towing flying machine.
(g) Sub-paragraphs (c) and (e) shall not apply to an aircraft flying under and in
accordance with the terms of a police air operator’s certificate.
(2) Converging
(a) Subject to the provisions of paragraphs (3) and (4), an aircraft in the air shall give
way to other converging aircraft as follows:
(i) flying machines shall give way to airships, gliders and balloons;
(ii) airships shall give way to gliders and balloons;
(iii) gliders shall give way to balloons.
(b) (i) Subject to the provisions of sub-paragraphs (a) and (b)(ii), when two aircraft
are converging in the air at approximately the same altitude, the aircraft
which has the other on its right shall give way.
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(ii) Mechanically driven aircraft shall give way to aircraft which are towing other
aircraft or objects.
(3) Approaching head-on
When two aircraft are approaching head-on or approximately so in the air and there is
danger of collision, each shall alter its course to the right.
(4) Overtaking
(a) Subject to sub-paragraph (b), an aircraft which is being overtaken in the air shall
have the right-of-way and the overtaking aircraft, whether climbing, descending
or in horizontal flight, shall keep out of the way of the other aircraft by altering
course to the right, and shall not cease to keep out of the way of the other aircraft
until that other aircraft has been passed and is clear, notwithstanding any change
in the relative positions of the two aircraft.
(b) A glider overtaking another glider in the United Kingdom may alter its course to
the right or to the left.
(5) Flight in the vicinity of an aerodrome
Without prejudice to the provisions of rule 39, a flying machine, glider or airship while
flying in the vicinity of what the commander of the aircraft knows or ought reasonably
to know to be an aerodrome or moving on an aerodrome, shall unless, in the case of
an aerodrome having an air traffic control unit that unit otherwise authorises:
(a) conform to the pattern of traffic formed by other aircraft intending to land at that
aerodrome, or keep clear of the airspace in which the pattern is formed; and
(b) make all turns to the left unless ground signals otherwise indicate.
(6) Order of landing
(a) An aircraft while landing or on final approach to land shall have the right-of-way
over other aircraft in flight or on the ground or water.
(b) (i) Subject to sub-paragraph (ii), in the case of two or more flying machines,
gliders or airships approaching any place for the purpose of landing, the
aircraft at the lower altitude shall have the right-of-way, but it shall not cut
in front of another aircraft which is on final approach to land or overtake that
aircraft.
(ii) (aa) When an air traffic control unit has communicated to any aircraft an
order of priority for landing, the aircraft shall approach to land in that
order.
(bb) When the commander of an aircraft is aware that another aircraft is
making an emergency landing, he shall give way to that aircraft, and at
night, notwithstanding that he may have received permission to land,
shall not attempt to land until he has received further permission so to
do.
(7) Landing and take-off
(a) A flying machine, glider or airship shall take-off and land in the direction indicated
by the ground signals or, if no such signals are displayed, into the wind, unless
good aviation practice demands otherwise.
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(b) A flying machine or glider shall not land on a runway at an aerodrome if the
runway is not clear of other aircraft unless, in the case of an aerodrome having
an air traffic control unit, that unit otherwise authorises.
(c) Where take-offs and landings are not confined to a runway:
(i) a flying machine or glider when landing shall leave clear on its left any
aircraft which has landed or is already landing or about to take off; if such a
flying machine or glider is about to turn it shall turn to the left after the
commander of the aircraft has satisfied himself that such action will not
interfere with other traffic movements; and
(ii) a flying machine about to take off shall take up position and manoeuvre in
such a way as to leave clear on its left any aircraft which has already taken
off or is about to take off.
(d) A flying machine after landing shall move clear of the landing area as soon as it
is possible to do so unless, in the case of an aerodrome having an air traffic
control unit, that unit otherwise authorises.