Hello Sacha Buchanan,
Can you please Insert below style on GD > Design > Scripts > Custom Style CSS and let us know how it goes.
.geodir-loc-bar .geodir-loc-bar-in .geodir-search .gd-search-input-wrapper {
width: 30% !important;
min-width: auto;
flex: 50 1 auto !important;
}
.geodir-loc-bar .geodir-loc-bar-in .geodir-search .gd-search-input-wrapper select{
-moz-appearance: none;
-webkit-appearance: none;
background-image: linear-gradient(45deg, transparent 50%, gray 50%), linear-gradient(135deg, gray 50%, transparent 50%), linear-gradient(to right, #ccc, #ccc);
background-position: calc(100% - 20px) calc(1em + 10px), calc(100% - 15px) calc(1em + 10px), calc(100% - 2.5em) 0.5em;
background-size: 5px 5px, 5px 5px, 1px 2.5em;
background-repeat: no-repeat;
padding: 0 10px !important;
cursor: pointer;
}