There is a reason that the home invasion rate is far higher in England than it is in the United States.
Firstly mate it's The United Kingdom
Secondly there is no Legal Definition of Home invasion in the United Kingdom.
However "burglary" in both countries is defined in a similar way as unauthorised entry to a property with the intent to commit a felony/crime, as are "homicide/murder" and "rape". So we could look at all three....
There were about 630K burglaries in the UK in 2012 (against 26M households) (ONS)
In the US, the FBI report lists 2.2M in 2009 (out of 114M households) (FBI).
Comparative rates: UK 1 per 41 households. USA 1 per 51 households. (USA dropped slightly overall since)
So you are around 20% more likely to be burgled in the UK, but hardly a massive difference.
However, how many burglaries are related to violent intent? "Assault" is very hard to compare as definitions vary (injury is not always a feature) but we do have stats for murders and rapes.
Murder UK total is 570 in 2012. (ONS)
Murder US total was 13,000 in 2009 (FBI)
Per 100K residents, that's 4 in the US vs. 0.9, so nearly 5X worse in the US. Again, the US rate dropped a few % but its still over 4.5X as high.
Similarly reported rape statistics per 100k for US and UK (2010) are 29 vs 28. Again no real difference (source UN).
So it does not seem very likely that "home invasions" where there is violent intent are higher in the UK. In fact I can't find any statistic from a reliable source (UK government, FBI etc) that would indicate anything of this nature.
So you are only slightly more likely to be burgled, but this does not seem to relate in any way to the chances that you will be murdered or raped, and I would guess the same is true of violent assault.