There are a number of assumptions to address in the question, so there are a variety of points to make in an answer.

Friction with Britain and the breakup of the Federation of Rhodesia and Nyasaland was in the early 1960s. The Unilateral Declaration of Independence was 1965, but the bush war didn't started to heat up until winter 1972, and military callups really got strong in 1976. I see them as different periods, and looking at the numbers, apparently so did the immigrants.

I wouldn't say that it could be called "so many people" without qualification. In the late 1950s, the average migration was some 14K per year, but in the early 1960s, it averaged 8K per year. The largest number of immigrants after UDI in 1965 was a sharp spike to about 15K in 1971. True, all these numbers were large in proportion to the white population, which was in the 200K range (finding housing and services for immigrants was a government concern), but of course not that large in absolute numbers. In 1973-4, it was down to 9Kish; there was a spike in 1975 to 12K, due to the collapse of the Portuguese empire; it trailed off down from 8K in 1976 to 3K in 1979.

As for who was coming, it wasn't just England. Yes, post-World War 2, there was a boom in British immigration, but it was a small majority of immigrants, and after that, South Africa's share increased and the UK's share decreased. Rhodesia relaxed or abandoned restrictions on immigration from the 1960s. More southern Europeans came in, and some Portuguese entered in 1975-6 with the collapse of their colonial rule (the final spike). Immigration and emigration details were to some extent secret, and to some extent records weren't kept, so I don't have great data for details. As for their attitudes, later in the 1970s, there were more criminals (after UDI and sanctions, there weren't Rhodesian immigration offices abroad to vet immigrants) and white nationalists (local chapters of the John Birch Society and the American Nazi Party starting in 1977).

Why immigrate to Rhodesia? The economy was a big factor. This was an advertisement in a Dublin newspaper (later 1960s, I'm guessing from context):

the weather ... If you want a spacious home, good wages, reasonable taxation, first-class school, expert medical attention, help for the missus in the house [meaning cheap black domestic servants] and a bright sunny future for all. If you want assisted passages for yourself and your family if you qualify, and a two year special concession, where a family with two children can earn up to £1748 (£200 Sterling) tax-free, ...

A Newsweek (U.S. magazine) ad in 1970 more focused on the economy. It wasn't a "bit better" living standard. In 1970, a sixth of all whites had a swimming pool. In 1952, Rhodesian whites had car ownership rates not seen in the U. S. until 1977.

Rhodesia was not the Hotel California -- you could get out without much trouble, even late when there could be issues with call-ups or financial penalties. Rhodesia had always had a significant percentage of immigration (I'm not sure non-whites were allowed in) and emigration, and people staying for only a few years -- from 1955 through 1979, immigration and emigration each averaged roughly 4% per year; over that period, there were more total immigrants and more total emigrants than the maximum population! Ian Smith was the only prime minister of Rhodesia (or Southern Rhodesia, as it was called before UDI) to be actually born in Rhodesia [1]. In 1971, about 1/4 of all whites in Rhodesia hadn't bothered to become citizens. Brownell writes,

The 1969 census revealed that three-quarters of Rhodesia’s white population over the age of 16 were born outside Rhodesia, of whom 59 per cent were born in either South Africa or Britain.... Of 1,460 people surveyed in the Rhodesian diaspora in the 1990s, only 36 per cent were born in Rhodesia, 39 per cent went to Rhodesia for jobs, and 22 per cent went as children.... In 1969, only 40 per cent of Rhodesians were born inside of Rhodesia. Among adults it was only 25.5 per cent.... In 1975, 45 per cent of the white population had been in Rhodesia for under ten years

The military callups: as I mentioned, the bush war was really starting in 1973, and ramped up in 1976. But in 1973 the callup was just 3,000, and half of them evaded it. In 1978, the Army said it needed at least 1K, but only 570 reported.

And until 1976, immigrants were exempt from conscription for 5 years. In that year, it was reduced to 2 years.

Summary: it was the economy, mostly. There was a long-standing culture of nipping in for a few years to make some money in a sunny land and then heading off somewhere else.

My major source is Josiah Brownell, The Collapse of Rhodesia: Population Demographics and the Politics of Race, I. B. Tauris, 2011. I've seen a review criticizing his conclusions, and his lack of opinions from Rhodesians and Zimbabweans, so I aimed to use mostly his primary quotations and figures. Barry M. Schutz, "European Population Patterns, Cultural Persistence, and Political Change in Rhodesia", Canadian Journal of African Studies, Vol. 7, No. 1 (1973), https://www.jstor.org/stable/483748, gave me some data for previous years.

[1] Using modern country names for the 8 prime ministers in order: South Africa, Botswana, UK, UK, New Zealand, British Embassy in Berlin, UK, Zimbabwe (Ian Smith).