True, and not just in the ways that you think
Does illegal immigration really change the electorate?
What is the time between an illegal immigrant enters the country and their gaining the right to vote in the U.S.?
It's quite difficult for illegal immigrants become a legal immigrants. However, the United States has the uncommon practice of birthright citizenship, which grants citizenship to anyone born within its borders. Hispanic birthrates have fallen recently, but still outpace the national average: 2.1, compared to 1.7.
So the answer to your question is either "never", or "one generation," depending on your perspective.
The most notable example is Texas which has long been a GOP stronghold, but many conjecture it will soon "turn blue" due to immigration (both legal and illegal).
Besides the problem of border control is there any difference between the Democratic Party and the Republican Party in political attitude towards the immigration laws which would prove the democrats are benefiting from illegal immigration as claimed?
The overwhelming majority of illegal immigrants are Hispanic, and Hispanic regularly vote Democrat. In 9 of the last 10 Presidential elections, Democratic candidates led by 20%+ among Hispanic voters.
And this behavior predates the current Republican emphasis on immigration laws. (While Republicans are typically tighter on immigration, both parties have mixed records -- recall that it was Republican icon Ronald Reagan that gave amnesty to millions of illegal immigrants in 1984.)
Race aside, immigrant voter demographics fit a Democratic strong spot: poor, urban, and young (44% of Hispanic votes are millennials, vs 31% nationally).
But illegal immigrants do not even have to vote or birth voters to change electorate dynamics. Two and a half million of California's 39 million residents are illegal immigrants.
Assuming these are represented in census reports (in theory all residents are included), California has three or four "extra" representatives just due to these illegal residents.
This is why the Trump administration argued to include a citizenship question on the general 2020 census.
The Trump administration argued in court that it needs to know the number and locations of noncitizens to better enforce the 1965 Voting Rights Act.
In 1964, the Supreme Court’s historic one-person-one-vote ruling ordered Alabama to draw legislative districts with equal populations, arguing that anything else made some people’s votes count less than their neighbors’. But including noncitizens in the calculus upends that, conservatives argue.
In Houston, redrawing maps without noncitizens could amount to a political sea change. Removing noncitizens from redistricting calculations would force immigrant-rich districts like Houston’s 29th to expand their borders to make up for the loss of population. Those districts are generally Democratic strongholds, and expanding them would often mean absorbing constituents from adjacent Republican areas.
(Trump was unsuccessful in his attempt.)
Note that illegal immigrants mostly live in urban areas, and urban areas vote strongly Democrat.
So regardless of what illegal immigrants' own political opinions are, their presence benefits the Democratic Party's voting power. It's a modern-day version of the Three-Fifths Clause: they can't technically vote but they still count.