1. Caveat
This is going to be an unsatisfactory answer, for the following reasons:
Boardman's claim is ambiguous: how are we to interpret the you in "you are statistically more likely to have a head injury walking"? A statistically average person in the country, considered over some period of time such as a year? Someone considering making a particular journey and wondering which mode is riskier? Or someone else?
I could find no official statistics which break down casualties by location of injury, so we're going to have to put up with "killed" and "killed or injured" as proxies for head injury figures in what follows.
These statistics seem likely to undercount injuries to pedestrians (relative to cyclists), because they only include road casualties that have been reported to the police. Many (perhaps most) injuries to pedestrians will not be reported as road casualties (either because they did not occur on a road, or because no vehicle was involved).
(No doubt you can think of more problems. Point them out in the comments, but please see if you can find relevant statistics. The difficulty here is not in spotting biases, but in finding data to address them.)
2. Absolute figures
In Great Britain in 2012, the average person was about 3.5 times as likely to be killed in a road traffic accident when walking as when cycling (and about 1.8 times as likely to be killed or seriously injured).
Source: Road Accident Statistics - Table RAS30001 - Reported road casualties by road user type and severity, Great Britain
2008 2009 2010 2011 2012
PEDESTRIANS
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Killed 572 500 405 453 420
Seriously injured 6,070 5,545 5,200 5,454 5,559
KSI 6,642 6,045 5,605 5,907 5,979
Slightly injured 21,840 20,842 20,240 20,291 19,239
All casualties 28,482 26,887 25,845 26,198 25,218
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PEDAL CYCLISTS
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Killed 115 104 111 107 118
Seriously injured 2,450 2,606 2,660 3,085 3,222
KSI 2,565 2,710 2,771 3,192 3,340
Slightly injured 13,732 14,354 14,414 16,023 15,751
All casualties 16,297 17,064 17,185 19,215 19,091
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3. By distance
In Great Britain in 2012, per kilometre travelled, the average person was about as likely to be killed in a road traffic accident when walking as when cycling, but about half as likely to be seriously injured.
Source: National Travel Survey - Table NTS0305 - Average distance travelled by mode, Great Britain.
2008 2009 2010 2011 2012
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Kilometres walked 311 316 294 301 290
Kilometres cycled 68 73 68 79 85
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PEDESTRIANS (per 100 million kilometres walked)
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Killed 3.0 2.6 2.2 2.5 2.4
Seriously injured 31.8 28.6 28.8 29.5 31.2
KSI 34.8 31.2 31.0 31.9 33.5
Slightly injured 114.4 107.6 112.0 109.7 107.9
All casualties 149.2 138.8 143.0 141.7 141.5
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PEDAL CYCLISTS (per 100 million kilometres cycled)
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Killed 2.8 2.3 2.6 2.2 2.3
Seriously injured 59.0 57.9 63.5 63.9 61.9
KSI 61.8 60.2 66.1 66.2 64.2
Slightly injured 330.9 319.1 343.9 332.1 302.8
All casualties 392.7 379.4 410.0 398.3 367.0
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4. Summary
Boardman is likely correct when considering a statistically average person in the country, and likely wrong when considering which transport mode to use for a particular journey. But I think his broader point is a fair one: the difference in casualty rates between the two modes of transport is not big enough to justify treating them differently in law.