Since the question and most of the answers were written, what had been strong recommendations from the government have been turned into regulations with the force of law, in the form of The Health Protection (Coronavirus, Restrictions) (England) Regulations 2020.
Of relevance to this question:
6.—(1) During the emergency period, no person may leave the place where they are living without reasonable excuse.
There then follows a list of reasonable excuses, which includes things like shopping for food, exercise, seeking medical assistance, providing care or assistance, going to work if it's not practical to work from home, and the like.
A trip, presumably for leisure purposes - whether with members of your household, or with one friend, or on your own - is not on the list, so would not be allowed.
Anyone breaking the regulations can be fined. The fines start at £60 (reduced to £30 if paid within 14 days), and then double for each subsequent fine, up to a maximum of £960.
The regulations (at least in their current form) expire on 26 September 2020.