You won' be going on your first solo until you are actually ready to do so, your FI will decide.
As for what happens, you will more than likely be out with your FI doing circuit consolidation, you will land and the FI will request from ATC over the RT for the student to be able to do their first solo. The ATCO will in 98% of the cases approve this, there may be times due traffic when a first solo simply isn't appropriate.
The FI will vacate the aircraft leaving you at the hold with you doing your checks before talking to ATC and letting them know you are ready.
ATC will have in the mean time informed the Fire Service that a First solo is due in the circuit.
Upon being ready for departure you will be given your circuit instruction from ATC and at an appropriate time cleared for take off. You will do one circuit and land.
You may have a short delay at the hold as we plan it so that you do not have to orbit once airborne , so it is a quick up and down.
A couple of things concern me about your worries. First things first there are plenty of stop bars at Cardiff, but they are not generally operated during the day, situations which require the operation of stop bars generally preclude the likelihood of a VFR flight for a first circuit.
St Athan is 2 1/2 miles away from Cardiff and if you read the rules you will be aware that it isn't a concern to you provided you remain in the circuit.
Windshear over the threshold at Cardiff, I doubt very much that your FI would put you up for a first solo if the wind conditions were such that turbulence could be expected.
Go back and talk to your FI, get them as necessary to take you on a trip of Cardiff ATC and let them explain what happens. Most importantly do not be disappointed if you do not go solo on your Birthday, it is more important that you are ready rather than aim for a particular date.