NASA generally doesn’t allow non-citizens to work there, which includes internships (not sure about the Rice Space Institute requirements). NASA has some intern opportunites with certain countries, but doesn’t look like UK is part of that, but of course OP can check into all of this. NASA International Internships - NASA
ETA: OP might consider applying thru QB as the family income seems in the ball park, assuming that’s $65K USD. Most QB partner schools allow international students to match, see here: https://www.questbridge.org/high-school-students/national-college-match/additional-citizenship-requirements