The UK does have one of the highest percentages but not the highest. So yes, the same phenomena is common to plenty of other developed nations, although "typical" might be to much of a strong word. The reasons are likely different for each one of them (see the plot at the end of the answer) but it does not seem to depend on the level of development (check Denmark or Italy by comparison with the UK, for example).
You can see the data in OECD.stat, more specifically in a query such as this (change variable to % of Foreign trained Doctors to reproduce the data in the following "table").
Country .............. % of Foreign trained Doctors
Israel..................... 57
New Zealand........ 42
Ireland.................. 41
Australia............... 32
UK........................ 28
USA...................... 24
Canada................ 23
France................. 10
Germany.............. 10
Austria.................. 5
Denmark............... 5
Italy....................... <1
I've tried to find some explanation to these numbers but even the more obvious seem to somehow elude a reasonable explanation. For example check this bar plot for the number of Medical Graduates per 100 000 people. Ireland having more than double the number of graduates, also as almost double the percentage of foreign trained doctors by comparison with the US (check the chart here).

EDIT: Apparently this subject has been widely studied in the EU.
Costigliola (2011), quoting the World Health Organization report "Health professional mobility and health systems. Evidence from 17 European countries (2011)" , argues that:
Among the most-cited factors for physicians’ mobility is the financial
motivation. As regards the salaries, major differences between
European countries can be observed. ...
Another factor influencing physicians to move across borders is the
working environment and conditions. The economic situation of a
country has a major impact on the quality and standards of healthcare
facilities and on the social benefits offered for health
professionals. ...
Training and career opportunities are also among the relevant
decision-making factors for physicians who consider leaving their
country of origin, either temporary or for a long period of time. ...
... and adds:
According to a WHO report on healthcare workforce migration in Europe,
there are also other factors associated with migration flows that can
stimulate migration and affect the choice of a destination country:
- Organizational factors, such as heavy workload, occupational risks, poor management, favouritism or lack of due process, lack of
recognition;
- Healthcare system factors, such as the absence or inadequacy of human resource policies, insufficient funding of health services, and
centralised decision-making;
- General environmental factors, such as poor economic conditions and lack of security.
Specifically for the UK the NHS seems to be in favor of professional mobility. I would also recommend the Politico.eu article "The EU exodus: When doctors and nurses follow the money" where this chart can be seen (for some countries it seems to somewhat contradict the OECD data):

As well as this one that show the sending vs the receiving country for these doctors (EU only):
