Personally I would say the correct way is Cost-a Coffee but I've heard some people use Coast-a.

Which is correct?

In all the (many) bits of the UK I know it's the same "o" as in "coffee", i.e. short. This is also how we pronounce the Spanish "costa" as in "Costa del Sol" or "Costa Rica". A shift to pronouncing it as "coast" seems unlikely in British English, especially with the extended alliteration (for want of a better term) in the coffee brand

Surprise, they’re [ˌɔə̯ɫkɚˈɻɛ̝t̚]!

Many, many other ac­tual pro­nun­ci­a­tions of this word can be
found among na­tive speak­ers of dif­fer­ent re­gional ac­cents.
Each one is “cor­rect’ for that ac­cent and speaker, even
should this dif­fer from one’s own.

Not only is English a pluri­cen­tric lan­guage
spo­ken na­tively by some 400 mil­lion peo­ple all across the globe,
even within a given ge­o­graph­i­cal re­gion, no­table pro­nun­ci­a­tion
dif­fer­ences abound be­tween that re­gion’s own nu­mer­ous in­ter­nal
di­alects.

There can there­fore be no sin­gle “cor­rect” pro­nun­ci­a­tion of
a word for all na­tive speak­ers ev­ery­where. This holds as true
of com­mon words like
all
and
right,
mother
and
daugh­ter,
warm and
bath,
as it does for less com­mon ones.

When you dis­cover ev­i­dence that na­tive speak­ers of dif­fer­ing
ori­gins, ed­u­ca­tions, or oc­ca­sions all say the same word
dif­fer­ently, that does not mean that one ac­cent’s pro­nun­ci­a­tion
for that word is “cor­rect” and the oth­ers are “in­cor­rect”. A
case in point is that you’ll never get ev­ery­one to say the word
cof­fee the same way.

If you don’t be­lieve me, just try to get folks to agree on what
the “cor­rect” pro­nun­ci­a­tion is of words like like route or
roof is. Each na­tive speaker will doubt­less say that their way
is “cor­rect”, but only those with the most severely provin­cial
of view­points would dare call al­ter­nate pro­nun­ci­a­tions from
other na­tive speak­ers “in­cor­rect”.

You might imag­ine that the per­ceived “owner” of a name has an
an­nointed right to de­ter­mine what is the “cor­rect” way to say
that name, but even here you can some­times be wrong. That’s
be­cause with proper names like those of peo­ple, cities, or
busi­nesses, this pic­ture be­comes even mud­dier. Anna, Charles,
Mel­bourne, and Ore­gon are all nat­u­rally pro­nounced dif­fer­ently
in some ac­cents than in oth­ers. You will never get a non-rhotic speaker
to say Charles the way its rhotic owner says it. Or just be­cause Anna her­self may
hap­pen to use the same vowel there as she uses in ap­ple does
not mean that some­one of a dif­fer­ent ac­cent who al­ways uses
the vowel from fa­ther for that word is some­how “in­cor­rect”.

Proper names orig­i­nally from a dif­fer­ent lan­guage than English
are some­times even more sub­ject to vari­a­tion, as these are not
nec­es­sar­ily only vari­a­tions by di­alect alone. Rather, ed­u­cated
na­tive English speak­ers with ex­pe­ri­ence in lan­guage the word
was bor­rowed may pro­nounce that word closer to how it is said in
the other lan­guage. Think of Paris, Ber­lin, Cairo, Louis­ville,
Lyons, Lima, New Or­leans, San­ti­ago, Co­lom­bia, Chile.

The par­tic­u­lar sound that seems to be con­cern­ing you here, the
stressed vowel from the first syl­la­ble of the word Costa,
hap­pens to be a sound that varies dra­mat­i­cally be­tween English’s
many re­gional di­alects.
This is fur­ther com­pli­cated by vary­ing im­pres­sions of how
as­sim­i­lated the word has been into English from its Ro­mance
roots.

It may sound just like the word coaster in some di­alects
but like the word caster in oth­ers. Some may rhyme it with how
they say pasta. It might even sound like the last two words in
the sen­tence “That’ll sure cost her”, or rhyme with the first
two words in “Lost a penny?”

Each ver­sion is “correct” in that ac­cent. That’s why all native speakers’ pronunciations of a word are “correct”.

Word pronunciations may vary depending on the broader phrase in which the word appears or on particular local historical factors. In California, where I live, a great many local place names are of Spanish origin. Often the standard native English pronunciations of these names indicate little awareness of Spanish pronunciation preferences—and yet these English pronunciations persist alongside pronunciations of the same or similar words or names that are not local and reflect a much stronger awareness of common Spanish pronunciation.

For example, whereas most (though not all) native English speakers in the San Francisco Bay Area pronounce the country name Costa Rica with a long o (KOAST-uh REEK-uh), and (back in the 1980s) many pronounced the anti-Sandinista Nicaraguan insurgency known as the Contras with a long o ("Koan-truhs"), they almost universally pronounce the East Bay county name Contra Costa with short o's ("KON-truh KOS-tuh").

Likewise, I have never heard a native English speaker in the Bay Area pronounce the last name Allende (in reference to Salvador or Isabel Allende) with an l sound rather than a y sound for the ll (the pronunciation is usually approximately "I-YEN-day"). Yet almost all native English speakers here pronounce the city name Vallejo with an l sound ("Vuh-LAY-ho"). (Note that they do use the Spanish-preferred h sound for the j.)

Further, most native English speakers here pronounce the familiar name of the artist Raphael in three syllables (approximately "Rof-I-EL" or "Raf-I-EL") but they pronounce the second word in the city name San Rafael in two syllables (approximately "Ruh-FEL").

The conclusion I draw from these (and similar) inconsistencies is that pronunciation is largely an aurally received phenomenon, and it is received from different sources at different times—rather than being formulated systematically and cross-checked consistently on the basis of identical or similar spelling. The name Costa Coffee is not one that I've encountered previously, so when I first met it in print (just now), I had no idea whether people who are familiar with it generally pronounce it with a long o or a short one. But the fact that people may be inclined to pronounce it different ways in different places doesn't surprise me at all.

Going back to the roots, the term Costa is an Italian surname which has nothing to do with Costa Rica or other coasts along which coffee plants are grown.
Costa in Italian is pronounced as cost - with a final a as in alpha. Any variant, elongation of vocals etc. are purely local interpretation of the term. (British, Americans or else)

Costa Coffee was founded in London in 1971 by the Costa family as a wholesale operation supplying roasted coffee to caterers and specialist Italian coffee shops.

Brothers Bruno and Sergio Costa founded a coffee roastery in Lambeth, London, in 1971, supplying local caterers. The family had moved to England from Parma, Italy, in the 1960s. Costa branched out to selling coffee in 1978, when its first store opened in Vauxhall Bridge Road, London.